George Gheverghese Joseph’s *Indian Mathematics: Engaging with the World from Ancient to Modern Times *is an accessible introduction to the history of Indian mathematics written in at a level appropriate for undergraduate mathematics students. The author also argues, as he does in his influential *The Crest of the Peacock: Non-European Roots of Mathematics*, for the inclusion of non-European mathematics in courses and in discussions of the development of mathematics. While the situation is better today than when *The Crest of the Peacock *first appeared in 1991, Joseph is still able to cite examples of treatises that seem to be unaware of mathematics outside the European tradition.

I will provide a more detailed summary of the portions of Joseph’s book that I expect are less familiar to those who teach an undergraduate course in the history of mathematics; I would expect many reading this review to already have some familiarity with the accomplishments and importance of Aryabhata I, Brahmagupta, Mahavira, Bhaskara II, and Ramanujan, for example.

Joseph begins this work with a discussion of some of the unresolved questions in the history of Indian mathematics. These include the term “Indian” in preference to “Hindu” or “Sanskritic”. Joseph’s decision to use “Indian” avoids present-day connotations of “Hindu” while at the same time acknowledging sources that are Buddhist, Jain, or Islamic. There was certainly also mathematics written in regional languages as well as in Persian and Arabic, so that “Sanskritic” is too limiting.

Joseph then turns to a set of problems he categorizes as the four “C’s”: Completeness, Continuity, Consensus, and Chronology. Joseph discusses each of these issues briefly and points out that one of the goals of his book is to restore continuity to the history of Indian mathematics. He then discusses the differing perspectives and evaluations of the quality of Indian mathematics and astronomy by those outside India, from around 1030 (Abu Rayhan al-Biruni) through 2009 (Kim Plofker). This first chapter also includes a handy chronology of Indian mathematics, taking us from the Harappan Period (roughly 3000 to 1500 BCE) through the Kerala Period (roughly 1300 to 1600 CE); Joseph’s book also includes mathematicians and mathematics that do not fit this chronology. I would also note here that Joseph has chosen to omit diacritical markings to simplify things for his readers; I will follow that same convention here.

In the second chapter, Joseph turns his attention to the place of Indian mathematics within the history of mathematics in general. “Indian mathematics from its very beginning saw its primary function as generating rules for systematic and efficient methods of calculation.” [15] This then is exhibited in the geometry of Vedic altars, the origins of Indian numerals, the close relationship between literacy and numeracy, and the appearance of zero in the Indian place value system and elsewhere in the world. Jain mathematics, with its relationship to Jain cosmology and astronomy, which is discussed briefly here, is followed by an introduction to the work of Aryabhata I (c. 500 CE). Quick surveys of early Indian algebra and contributions of the Indian mathematicians to trigonometry through the time of Bhaskara II (c. 1150 CE) complete the introduction to Indian mathematical contributions. This sets the stage for Joseph’s analysis of global interactions in mathematics among the cultures of India, Egypt, Mesopotamia, Greece, the Islamic world, and China. He proposes many possible interconnections, while admitting that some connections are unconfirmed, that is, the evidence for some is not available.

Chapter 3 turns its attention first to a discussion of mathematics in the Harappan civilization. Joseph details the discovery of this civilization in the early twentieth century. No one has yet deciphered the Indus script; Joseph discusses attempts to do so. Without a written record, we are left to consider material evidence from the culture, categorized by Joseph as the cities themselves and their monuments, artifacts of everyday use, a few pieces of sculpture, the remains of the fauna and flora, a number of seals made from steatite or terracotta, and a few remains of human skeletons. Plumb-bobs of the culture show an interesting set of ratios that are a mixture of binary and decimal. Scales and rulers for measuring length have been uncovered. Bricks may be of different sizes, but have a common shape with dimensions in the ratio 4:2:1. Joseph then considers the mathematics of the Vedic Age (roughly 1500 to 500 BCE). The written record of geometrical knowledge for the construction of altars meeting specific conditions occurred as early as 1000 BCE; these were also found later in the *Sulbasutras *(800 to 200 BCE). Joseph suggests that the *Sulbasutras *include what may be regarded as the first applied geometry text. An appendix to this chapter provides details of some of the constructions and transformations that used a version of the Pythagorean Theorem to transform a rectangle into a square of equal area and, necessarily only approximately, a circle to a square of equal area, among other interesting constructions. A rational approximation for \(\sqrt2\) as \( 1 + 1/3 + 1/(3\times4) - 1/(3\times4\times34)\) is also presented and discussed.

The mathematics of the Jain and Buddhist period (roughly 300 BCE to 400 CE) is the subject of the fourth chapter; Joseph explores a number of mathematical topics. First is the Indian “obsession” with large numbers (up to the 421st power of 10 in a Buddhist work *Lalitavistra* and an “ultimate” number of metaphysical significance in Jain texts, namely 8400000 to the 28th power!). He then offers a discussion of the evolution of the Indian decimal place value system, including a history of zero and its dissemination to the east and west. The Jains developed a concept of infinity, which Joseph points out was “not mathematically precise, [but] was by no means simple.” [125] The classification of numbers was complex indeed, as enumerable (lowest, intermediate, and highest), innumerable (nearly innumerable, truly innumerable, and innumerably innumerable), and infinite (nearly infinite, truly infinite, and infinitely infinite). Further, the Jains recognized different kinds of infinity (in one direction, in two directions, in area, perpetually, etc.) and of different sizes. Infinitesimals also appear. Other considerations of the cosmological structures led to developments in sequences and progressions, exponents and logarithms, and permutations and combinations. Joseph mentions that the notable contributions of Jain geometry were in measurements associated with circles — computational formulas for the circumference, diameter, area, chords, versines, and arcs of segments.

The principal subject of Chapter 5 is the Bakhshali Manuscript. It was literally dug up only in 1881; the dates of the document and of its contents (for it may well be a copy of earlier work) are a matter of dispute. Suggestions of scholars range from the fourth to the twelfth centuries CE. It is possibly the first work of Indian mathematics free of religious or metaphysical considerations. It contains only a few problems in geometry; rather it is devoted to procedures and examples in arithmetic and algebra. Topics include procedures for working with fractions, extracting square roots, computing profit and loss, and calculating interest, the Rule of Three, arithmetic and geometric progressions, simultaneous equations, indeterminate equations, and quadratic equations. Joseph concludes the chapter with a brief discussion of the importance of Indian astronomy, which developed in parallel with the mathematics, and a summary of Indian mathematics through 500 CE. At that time begins the classical age of Indian mathematics.

I feel that much of the story that Joseph presents in the next few chapters is more familiar to those teaching undergraduate history of mathematics courses, though certainly many of the details may not be so familiar. In any case, my summary here will be brief. Aryabhata I and his school are discussed in Chapter 6 through a summary of the *Aryabhatiya *(499) and commentaries on it. The discussion of the mathematics of the *Aryabhatiya* is useful; Joseph has chosen, sensibly for the sake of his audience, to omit discussion of the astronomy in this and later works. Joseph includes also a discussion of Bhaskara I, who, Joseph notes, does not receive the attention he deserves. Chapter 7 begins with Brahmagupta and his *Brahmasphutsiddhanta* (628), then considers Mahavira and his *Ganitasarasangraha *(c. 850). In Chapter 8, Joseph discusses Bhaskara II and his *Sidhantasioromani *(1150). Of its four parts, two are on astronomy and two are on mathematics: the *Lilavati *(on arithmetic) and the *Bijaganita *(on algebra). Joseph also remarks that it is in one of the parts on astronomy, the *Ganitadhyaya* (on the celestial globe), that Bhaskara uses the concept of instantaneous motion and an infinitesimal increment.

In Chapter 9, Joseph returns to topics that may be less familiar, beginning with the work of Narayana. In addition to his arithmetical treatise *Ganitakaumudi *(1356), there is also a partially extant work on algebra *Bijaganitavatamsa* and a commentary on *Lilavati.* The *Ganitakaumudi* is notable for unusual topics such as writing fractions as the sum of unit fractions, finding factors of integers, casting out nines (or other numbers) to check computations, and constructing magic squares. This chapter concludes with a brief section on a school of mathematicians and astronomers from the sixteenth and seventeenth centuries.

Chapter 10 is devoted to the school of mathematicians in Kerala in southeastern India. The mathematics of Kerala is of particular interest, as the scholars there developed numerical integration methods and infinite series derivations for some trigonometric functions and for the computation of \(\pi\) some 250 years before calculus was developed in Europe. Joseph begins by discussing that, in the history of the history of mathematics, relevant developments in Indian mathematics after Bhaskara were generally not understood in Europe until the second half of the twentieth century, and even then were omitted from some texts on the history of calculus. Joseph provides a schematic diagram of some thirteen mathematicians in Kerala who are of interest, two from the ninth century and eleven from between the fourteenth and nineteenth centuries. As simply one example that Joseph offers is Nilakantha’s fifteenth-century result on the power series for sine and cosine. Besides presenting the mathematical results from this school, Joseph provides background on the society of Kerala and the possible methods and motivations of the mathematicians. He also discusses the possibility of transmission of the Kerala results to Europe in the sixteenth or early seventeenth centuries. The appendices to this chapter discuss the Madhava-Gregory series, namely that \[\frac{\pi}{4}=1-\frac13+\frac15-\frac17+\dots,\] transformations to improve the accuracy and rate of convergence of this series, and the derivation of the power series for sine and cosine.

In Chapter 11, Joseph offers a history of trigonometry in India from Aryabhata through Nilakantha. I found the result of Bhaskara I that offers a rational function approximation for the sine function for arcs between 0 and 180 degrees to be truly remarkable. Simply compare the graphs or tabular values of \[y=\sin(x)\qquad\qquad\text{ and }\qquad\qquad y=\frac{4x(180-x)}{(40500-x(180-x))}\] for \(x\) in degrees.

Chapter 12 discusses some of the mathematics from sources that were not in Sanskrit. Joseph notes that the Jain and Buddhist material used the Prakrit language, that the Bakhshali Manusrcript was written in Sarada, and that some of the Kerala mathematicians wrote in Mayalayam. This chapter considers first a work by Thakkura Pheru (1315, written in Apabhramsa). Joseph then turns to Pavuluri Mallana to discuss an eleventh-century translation of a work into Telugu; this work contained examples of “necklace numbers” producing patterns such as such as \(111111 \times 111111 = 12345654321\) and \(146053847 \times 338 = 65432123456\). A brief discussion on recreational mathematics follows. The chapter concludes with a section on Indian mathematics within the Persian-Arabic tradition.

Joseph’s book concludes with Chapter 13, which discusses the interactions between western European mathematics and the Indian tradition. The evaluation and translation of Indian mathematics by Europeans is a story of its own, beginning with the French astronomer Guillaume Le Gentil in India to observe the transit of Venus in 1769. Some Europeans, such as Reuben Burrow (1783), reacted with wonder and respect. Others, such as John Playfair (1789), chose to explain accurate ancient Indian astronomical data by conceding great antiquity in careful measurements rather than acknowledging the analytic developments of Indian mathematics. Still others, such as Thomas Macaulay in 1835, denigrated the Indian accomplishments, writing, “But when we pass from works of imagination to works in which facts are recorded and general principles investigated, the superiority of the Europeans becomes absolutely immeasurable.” [447]

Charles Whish’s 1832 paper in which he reported that Nilakantha’s work “laid the foundation for a complete system of fluxions” [448] was largely ignored. The final two Indian mathematicians that Joseph discusses are Yesudas Ramchandra and Srinivasi Ramanujan. Ramchandra’s *A Treatise on the Problem of Maxima and Minima Solved by Algebra *(1850) can be seen “bridging the discontinuity between an Indian mathematical tradition ... and modern calculus.” [450–451] Championed by Augustus De Morgan in England, it failed to provide a pedagogical bridge between the disappearing tradition of the native Indian algebraic mathematics and the colonial western European mathematics in schools in India. Ramanujan’s genius and life need no particular explanation here.

Joseph’s last section provides a summary of the Indian “Way of Knowing” mathematics. “To conclude, the roots of the ‘unreasonable effectiveness of mathematics’ that provided the engine for the Scientific Revolution are found in the combination of Greek rational and deductive traditions and Indian (Chinese) algorithmic/empirical traditions. Such a combination was mediated through the agency of the Islamic conduit.” [461]

There are some unfortunate typesetting errors that have crept into the book. Some of these errors might cause difficulty for attentive undergraduate readers, but most are trivial.

This book is very useful. It will join Kim Plofker’s *Mathematics of India *and her portion of Victor Katz’s *The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook* on my bookshelf as a primary reference for teaching the section on India in my course on the history of mathematics.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.