BOOK I REVIEW Indian Mathematics and zero-symbol, namely a dot (.), was used' Astronomy: Some Land­ in Pingals's Chandal;sutra, but this state­ ment is not based on textual evidence. marks At least we should distinguish zero as a concept from zero as a mathemati­ Michio Yano* cal symbol. The author follows the old view of dating the B akhshali M anu­ Indian Mathematics andAstronomy: script in the 3rd century A.D., but the Some Landmarks·· reader is advised to refer to the most By S Balachandra Roo Jnana Deep Publications, Bangalore, recent and convincing result of Prof. T 1994, Pages, VIII + 234, Price Rs. 751- Hayashi in his Bakhshiili Manuscript, On Indio). edited with an English translation and commentary, Groningen, 1995. This book is 'mainly addressed to the student community and general read­ The main part of this book is divided ers, with a view to providing an intro­ into 11 chapters. The author's attitude duction to the development of mathe­ is quite fair being free from nationalis­ matics and astronomy in India', as is tic bias, which have been often embar­ announced in the Preface. The present rassing to the present reviewer. The reviewer congratulates Prof. Rao on reader can get considerably fair evalua­ his successfully achieving this aim. The tion of the Indian contribution to math­ scarcity of books of this kind seems to ematics. But there still remain some have deprived the young Indian gener­ views which might misguide the reader. ation of the chance of being informed Let us review them following the order of their own scientific heritage .. In this of the chapters. very compact book of about 200 pages Chapter 1 outlines the mathematics of the author has packed almost all the es­ the Sulvasutras. The interesting topics sential aspects of the history of Indian of square-roots and squaring the circle mathematics and astronomy from the are dealt with. Commenting on the cor­ Vedic period to Ramanujan. rectness of the value of v'2 the author In the introduction entitled as 'Chap­ proposes to :rename' the 'Pythagoras ter 0' the author gives an overview of theorem as Sulva Theorem'. It seems ancient Indian mathematics. Most top­ that the author did not remember the ics in this chapter are taken up more in fact that the similar or even better val­ detail in the following chapters. Here ues were attested in the Old Babylonian are two comments of the reviewer on cuneiform texts belonging to the time the topics which are found only in this long before the SUlvasiitras. chapter. The author claims that 'the Chapter 2 summarizes the Vediingajyo- *Reproduced from Ganita Bharati (edited by R C Gupta), Vol. 19, pp. 128-132, 1997. ** A second and revised edition has :en brought out in 2000. ~______________ _ RESONANCE I July 2000 BOOK I RtVltW ti~a. According to the author, Lagad­ Kerala where Aryabhata was highly es­ ha's Vedangajyoti$a is assigned to 'the teemed and where his school found a period of 12th to 14th century B. C.', remarkable revival in Madhava school. based on the position of the winter sol­ As the starting point of this long and stice which was 23°20' apart from that extraordinary tradition, the author's st­ in the day of Varahamihira. In dIs­ ress on Bhaskara 1's significant position cussing such matters one should be care­ is truly appreciated. ful of the fact that the origin of a myth Chapter 5 describes the works ofVaraha­ and origin of the texts based on the mihira. The author follows the old tra­ myth are quite different. Any myth dition that Varahamihira was a 'Maga­ could go back to the time immemorial, dha-dvija' and that he migrated from as some people like to say, but texts Magadha to UjjainI where he was ac­ cannot. It is true, for instance, that tive as a court astrologer. But there is KJ;ttika(Pleiades) was near the vernal a view that 'magadha' was a result of equinox in 2300 B.C., and thus it is misinterpretation of the original form not wrong to think that the origin of 'mag a', which stands for the Zoroas­ such an idea goes back to the Indus Val­ trian priesthood of Iranian origin. There ley Civilization, but this does not mean are several evidences to support this that already at that time the ecliptic view, especially in the Paiicasiddhantika, coordinate system was well defined. where Iranian deities were associated In Chapter 3 the author focuses his at­ with the days in a month. The rest of tention to Aryabhata I, summarizes the Prof. Rao's accounts on Varahamihira's contents of the Aryabhatlya, and dis­ achievement, including his indebtedness cusses his date and place of activity, to Greek astrology, is well founded and and his innovative contributions in ma­ guides the reader to the right direction. thematics and astronomy. It is true Chapter 6 dwells on Brahmagupta, es­ that Aryabhata used the word asanna pecially on his controversial character. ('approximately') for his excellent value His critical views on the established au­ 1f= 62832/20000 but there is no evi­ thorities, which are often very severe, dence to show that 7r was regarded as are neatly summarized in this chapter. irrational by Aryabhata himself. The author also quotes the words of Chapter 4 is devoted to Bhaskara I, who AI-BlrunI who was one of the earliest is the author of the oldest extant com­ ancient scholars who could offer a fair mentary on the Aryabhatlya. As Prof. evaluation of Brahmagupta's contribu­ Rao points out, it is interesting to note tion. The reader is introd uced to the fa­ that Bhaskara I went into oblivion in mous Brahmagupta's theorem on cyclic north .India, and that his works sur­ quadrilateral and his ingenious solution vived only in the south, especially in of the indeterminate equation of the sec- -RE-S-O-N-A-N-CE--I-J-UI-Y-2-0-00-------------~-----------------------------8-9 BOOK REVIEW ond degree ( vargaprak'(ti). Brahmagup­ lish translation the charming examples ta's most outstanding contribution, the of mathematics given in the Lilavati. second order interpolation, is also ex­ plained in a clear manner. Chapter 9 is devoted to GaI).esadaivajiia who is one of the best commentators of Chapter 7 deals with Mahavlra, one of Bhaskara II's works. A quotation from the most renowned Jain mathematicians Ga~esa's father Kesava is quite inter­ belonging to the 9th century. The his­ esting in that he criticizes the tradi­ tory of Jain mathematics is one of the tional Siddhantas and stresses the im­ most neglected chapters of Indian math­ portance of observation and modifica­ ematics and therefore further study is tion of parameters. Prof. Rao explains strongly expected. I hope this chapter Ga~esa's interesting way of giving the stimulates the young scholars' interest date of composition of the Vrndavana­ in this unexplored field. The author re­ t[ka. The resulting date is' Maghasukla­ gards that the 'ayatav'(tta' (lit. oblong ~ratipad of Saka 1500'. Prof. Rao re­ circle), treated in the Ganitasarasa'fllg­ gards this date as falling in A.D. 1578, raha, is identical to 'ellipse', but in fact, but this is wrong. According to the they are different in the strict sense of present reviewer's computer program ba­ modern mathematics. sed on the Suryasiddhanta, the date cor­ Chapter 8 makes a good introduction to responds to January 27 (Tuesday), A.D. the achievements of Bhaskara II, whose 1579. Such kind of error (in this case popularity outstrips all other Indian ma­ the author simply added 78 to 1500 with­ thematicians. The author is correct wh­ out taking into account the month) is en he says that Bhaskara II was the frequently committed when Indian date first Indian astronomer who took into is converted into the Western date or account the effect of evection in lunar vice versa.- equation, but to say that 'Bhaskara's Ga~esadaivajiia's Grahalaghava gave the discovery preceded this in the west (by most significant influence on the Pafican­ Tycho Brahe) by nearly four centuries' ga makers. It is interesting to know is wrong, since Ptolemy in the second that even today many popular calen­ century A.D. already knew the second dars are based on this work. anomaly' of the moon which was caused byevection. Among several innovations Chapter 10 forms the most important ascribed to Bhaskara II, the cakravala part of the entire book, giving consider­ method of solving the indeterminate eq­ ably detailed accounts on the outstand­ uation of the second degree and the con­ ing activities of the astronomers in Ker­ cept of differentials are skillfully explai­ ala. The author rightly criticizes the ig­ ned in this chapter. The author at­ norant view of some famous historians tracts the reader by quoting with Eng- of mathematics that there were no im- -90-----------------------------~-------------R-E-S-O-N-A-N-CE--I-J-UI-Y--2-00-0 BOOK I REVIEW portant mathematicians after Bhaskara ter the author adds a brief section 'He­ II, although the high achievements of liocentric model in Kerala astronomy', south Indian astronomer-mathematician which is based on the recent paper of had been brought to light by C M Whish Ramasubrahmanian et al. It is true as early as in A.D. 1835. that NIlaka1;1tha tried to construct a new geometrical model in which the effects The author introduces the reader to the works of Paramesvara, Damodara, NIla­ of two epicycles, i.e. mandavrtta and sighrav'(tta be represented in a unified kalftha, Jye1?thadeva, Sankara, Naraya­ form, but the result cannot be called ~a and Acyuta.