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Mathematics in India.

Mathematics in India. By KIM PLOFKER. Princeton: PRINCETON UNIVERSITY PRESS, 2009. Pp. xiii + 357.

Historians of mathematics encounter many hurdles when they try to understand the development of mathematics in India. The texts are composed in a highly compressed form in Sanskrit verse and very few are available in modern English translation. In many cases, the chronology of texts is still unsettled. Moreover, mathematics was not an independent discipline (sastra); it developed largely in the context of astronomy. Even the term ganita is not exclusive to mathematics, for it denotes mathematical astronomy as well. What is more, ganaka, the possible term to designate a mathematician, is more frequently used for astronomers and astrologers.

Kim Plofker's meticulously researched and engagingly written Mathematics in India aims to remove these hurdles and to present the development of mathematics "as a coherent and largely continuous intellectual tradition, rather than a collection of achievements to be measured against the mathematics of other cultures," as was done in earlier histories of Indian mathematics. Though it follows the mainstream narrative, the strong point of the book is that it endeavors to present other points of view quite objectively and admits it whenever there is a lack of supporting evidence for the mainstream narrative. It provides the proper historical and intellectual context to the texts, and introduces the main contents through copious extracts that are impeccably rendered into English and lucidly explained in modern notation.

The narration begins in Chapter 2 with the mathematical thought in the Vedic texts. There are three strands of this thought. First, the Vedic hymns are replete with specific terms to designate higher powers of ten that go far beyond the requirements of computation in daily life. There are also other significant references to numbers and their properties. Second, the rules prescribed in the sulva-sutras for the construction of different types of sacrificial altars involve area-preserving transformations of plane figures, from square to rectangle or to circle and so on. In this context, it is postulated that the squares on the width and length of any rectangle add up to the square on its diagonal, which is otherwise known as the Pythagorean theorem. Third, the Jyotisa-vedanga, which belongs to the post-Vedic, but pre-Classical, Sanskrit corpus and which was composed somewhere around the fifth or fourth century B.C.E., provides "the first available link between the ambiguous celestial and calendric utterances of the Vedas and the full-blown Sanskrit mathematical astronomy of the first millennium C.E."

Chapter 3 deals with the "mathematical ideas and methods during the centuries just before and just after the turn of our Common Era." Important clues are provided by three texts, viz., Artha-sastra, Yavana-jataka, and Panca-siddhantika. The Artha-sastra envisages a large numerate bureaucracy, trained in measurement and quantification. Though the inscriptions of this period contain non-place value numbers, there are allusions to place value in literature. The earliest occurrence of decimal place value notation is in the third century C.E;., in the Yavana-jataka, where it was stated that the work was composed in the year "visnu (1)-hook-sign (9)-moon (1)," i.e., 191 of the Saka era, which translates to 269 C.E. The Yavana-jataka contains an Indianized version of traditional Greek astrology, together with the twelve signs of the Greco-Babylonian zodiac. The fusion of the luni-solar astronomy of Jyotisa-vedanga with its circle of naksatra constellations and the Mesopotamian-influenced computational schemes of Greek astrology laid the foundation for all subsequent developments in Indian astronomy. The Panca-siddhantika also shows a similar blend of Indian numerical concepts and methods derived from Mesopotamian spherical astronomy, such as terrestrial latitude and longitude on a spherical earth, and celestial latitude and declination for heavenly bodies. It shows furthermore how Indian astronomers modified the rather clumsy Greek geometry of chords with the simple geometry of Sines.

Since mathematics developed in India in the context of astronomy, Chapter 4 is devoted to mathematical astronomy. The chapter begins with an introduction to geocentric astronomy, which is followed by a fine discussion of the evolution and nature of siddhanta astronomy, with long extracts from Brahmagupta's Brahmasphuta-siddhanta and Lalla's Sisya-dhi-vrddhida. The parameters of the siddhantas were originally borrowed from Hellenistic Greek sources and then modified in an ad hoc way to conform to isolated observations. The main features of the siddhantas are (i) the use of large integer parameters to calculate mean positions of celestial bodies, (ii) the use of trigonometry to calculate true positions in orbits involving eccentric or epicyclic circles, (iii) mathematical prediction of significant planetary positions, and (iv) computation of eclipses. For expressing large numbers within the metrical form of Sanskrit verse, different strategies were developed, using either single consonants or significant words to denote the numbers 1 to 9 and zero. The former was limited to South India, mainly to works composed in Kerala.

After this detailed analysis of the astronomical context, Chapter 5 finally addresses itself to mathematics proper and looks first at the chapters on mathematics in astronomical siddhantas, viz., Aryabhata's Aryabhatiya and Brahmagupta's Brahmasphuta-siddhanta, and then at the early independent texts, viz., the Bakhshali Manuscript and Mahavira's Ganitasara-sangraha, with detailed analysis of their structure and contents, illustrated with long extracts. It is unfortunate that Sridhara's Pati-ganita and Sripati's Ganita-tilaka are ignored in this treatment, though they predate the Ganitasara-sangraha as independent texts on mathematics. In particular, Sridhara is probably the first to write an independent work on mathematics, at least a century before Mahavira. He also seems to be the first to have written a separate book on algebra, which is now lost, but was known to Bhaskara II. Moreover, the fact that Sridhara wrote first a larger work of which he later prepared an abridgement shows that there existed a circle of readers distinct from the readers of astronomical works. The Bakhslali Manuscript, which appears to be a digest of extracts from other works, also indicates a distinct readership for these mathematical texts.

The mathematical texts discussed above share many results and methods, but they also display wide divergence in formal and content. Some kind of standardization in this regard was achieved in the twelfth century in the works of Bhaskara II, which are described in Chapter 6. Bhaskara II composed Siddhanta-siromani. Lilavati, and Bija-ganita on mathematical astronomy, arithmetic, and algebra respectively. These became extremely popular and attained a near-canonical status. Citing an inscription issued by a descendant of Bhaskara in 1207 commemorating the establishment of a college for the study and propagation of Bhaskara's works, the author concludes that "Bhaskara thus possessed the advantage of a renowned scholarly lineage and a tradition of royal and noble patronage. ... [T]hese advantages were doubtless partly responsible for the successful dissemination and lasting popularity of his works, but other factors were involved as well." It is difficult to agree with this conclusion. For one thing, Bhaskara himself was not associated with any royal court. Nor do any of his commentators praise him for his lineage or for royal patronage. The reasons for his popularity must, be sought in the works themselves. These reasons are, as the author herself admits, the "careful organization of the works and their clear exposition." The high poetic quality of his works must, also have played some role in making the works popular. Bhaskara composed a commentary on his Siddhanta-siromani, to which he gave the significant name vasana-bhasya, thus laying stress on the role of logical demonstration or rationale in mathematical discourse. In this context, Plofker makes a perceptive observation on the nature of proof in the Indian mathematical tradition and how it differed from the definition-proposition-proof system of the Euclidean tradition. Logical demonstration in Sanskrit texts, she says, "was an individual mathematician's ingenuity rather than a formal methodology on which he had to rely for perceiving a mathematical fact in its different guises--verbal, numerical, symbolic, or geometric." "And it was this act of perception, this seeing a mathematical relationship 'like a fruit in the hand,' that the mathematician strove for, instead of a mechanical sequence of approved steps leading to a logically unassailable conclusion."

While the rest of India generally pursued Bhaskara's works and elaborated upon them, an entirely novel path was trodden in Kerala by Madhava and his pupils in the fifteenth and sixteenth centuries, which is the subject of Chapter 7. Interestingly enough, in Kerala mathematical astronomy (and probably also other disciplines of Sanskrit scholarship) was cultivated not only by Namputhiri Brahmins but also by lower castes like the Ambalavasis. Interestingly also, original works on mathematical astronomy were composed here for the first time in Malayalam prose as against the general practice of employing Sanskrit verse for scholarly discourse. The highpoint of the Kerala school is the infinite series for trigonometric quantities discovered by its founder Madhava, such as the derivation of the Madhava-Leibnitz series for [pi] and the Madhava-Newton power series for the Sine and Cosine and so on. Though Madhava's work continued to be studied and elaborated upon by an unbroken succession of pupils, it is intriguing that his most important works on the power series do not survive in the original, but only as citations in the works of his pupils and their pupils. Madhava's school treated explanations and rationales as valid subjects for mathematical creativity in their own right. The commentary of Nilakantha on the ganita chapter of the Aryabhatiya provides detailed rationales for all its rules, many of them geometrically very ingenious. The most comprehensive of these works on rationales is Jyesthadeva's Malayalam work Yukti-bhasa which explains and demonstrates the mathematics in Nilakantha's Tantra-sangraha, from the eight fundamental operations of arithmetic up to the spherical geometry of the moon's phases. Other notable achievements of the Kerala school are Paramesvara's eclipse observations for a period of over fifty years, which he recorded in his Siddhanta-dipika, Nilakantha's bold assertion that a siddhanta which does not conform to observation should be discarded; his attempt to resolve the computational inconsistencies in the earlier models by making the star-planet's actual revolutions no longer geocentric, and so on. Plofker is of the opinion that there is no evidence that any of these innovations were derived from the Greco-Islamic mathematical science, nor for the hypothesis that the Kerala school inspired the development of calculus in Europe.

Chapter 8 examines the exchanges with the Islamic west. In the eighth and ninth centuries, the Islamic world avidly received Indian decimal place numerals, arithmetic, and trigonometric functions. But, after about the early tenth century, the Islamic world exclusively followed the Hellenistic tradition. Exchanges continued sporadically in India, but did not leave any significant impact on the development of Sanskrit astronomy or mathematics. At the courts of Muslim kings and elsewhere, some Islamic works were rendered into Sanskrit, such as Mahendra Suri's Sanskrit manual on the astrolabe or the anonymous Hayata-grantha, and some Sanskrit texts like the Lilavati were translated into Persian. More important was the naturalization of certain forms of Islamic astrology in Sanskrit under the name tajika. Likewise the zij-style of astronomical tables inspired the production of extensive Sanskrit tables called kosthakas. While astrolabes and tables had a significant impact on the practice of Sanskrit astronomy, the axiomatic deductive geometry pursued by Islamic mathematicians seems to have sparked no similar imitations in Sanskrit. What is not expressly stated but is relevant here is the fact that Sanskrit mathematics remained untouched by the mathematical achievements of the Arabs, from al-Khwarizmi onwards.

The ninth and final chapter deals with the status of astronomy and mathematics during 1500-1800 and describes briefly the lives of individual scholars, their family and academic lineages, centers of study, and patterns of migration from small towns in Maharashtra to Varanasi and/or to the Mughal court. There are no notable additions to mathematical literature, but dozens of Sanskrit table texts were produced between the late fifteenth and the late eighteenth centuries, suggesting that a significant portion of the mathematical interest and energy of early modern Indian astronomers went into devising methods for computing tables and for using them to generate yearly calendars and predictions of planetary positions at desired times. The idea of detailed demonstration seems to have gripped more and more mathematical imagination in the sixteenth century, as can seen from the commentaries on the Lilavati by Ganes'a and on the Bija-ganita by Krsna.

Appendix A offers a useful introduction to the basic features of the Sanskrit language, structure of the Sanskrit verse, and related matters. Appendix B contains biographical data of some forty Indian mathematicians, who are, as can be expected, mostly astronomers. An extensively bibliography and index complete the volume.

Plofker's attempt to situate Indian mathematics in the proper context led to a very detailed treatment of mathematical astronomy (at times far more detailed than mathematics itself), but that is no drawback, for the book serves as an excellent introduction to mathematical astronomy as well.

In the concluding verse of his Bija-ganita, the great Bhaskara advertises his book in these words: 'To widen knowledge and to attain depth, read, do read, mathematician, this small volume, elegant in style, easily comprehensible even to the uninitiated, dealing with the essence of entire mathematics, and containing the demonstration of its principles, replete with excellences and devoid of any defects" (Colebrooke's translation, slightly altered). In recommending this book to historians of astronomy and mathematics and also to Indologists in general, we can do no better than to repeat Bhaskara's words.

S. R. SARMA

DUSSELDORF
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Author:Sarma, S.R.
Publication:The Journal of the American Oriental Society
Article Type:Book review
Date:Jan 1, 2010
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