Gaṇitānanda

Preparing to write this review, the reviewer had a thought that the book offers an experience somewhat akin to listening, over a period, to a grandparent narrating tales from the past. And why not? The book is, after all, an 80th birthday commemoration consisting of selected papers of a historian, who devoted most of his working life to popularizing the history of mathematics in ancient India, published over 500 articles in the area, edited a journal of history of mathematics–-Gaṇita Bhāratī–-for over a quarter-century, had intense interactions with the world community of historians of mathematics as a member of the International Commission for History of Mathematics (ICHM), and was honoured with the highest prize in the area of history of mathematics, the Kenneth O. May prize: Professor Radha Charan Gupta.

A few points about the author, Radha Charan Gupta (RCG for short, in the sequel), and his work would be in order, at the outset, to place the book in perspective. It is a pleasure to note that by and large, the work follows an approach driven by an inspiration to explore the truth as it may be, with no a priori preference, and the attitude well approximates the scientific ideal of objectivity. Apparently, the rather negative review of the book History of Hindu Mathematics by B.B. Datta and A.N. Singh in Mathematical Reviews, in 1963, was one of the “events” that made RCG “curious and restless”, leading him to take up the study of history of mathematics and “played a crucial role” in shaping him, as he put it.¹ And yet, in his article “Foreign Reviews and Evaluation of Indian Works in History of Science” (gb-1989; II.3),² he may be seen to recount the criticisms by foreign reviewers of numerous Indian works, including the above-mentioned review, without rancour or hostility towards the critics or the critical comments. Some of the criticisms he is seen to endorse, while others are left uncommented, suggestive of an open mind. He observes (page 22):

The book offers an experience somewhat akin to listening to a grandparent narrating tales from the past

More instances of poor quality work on the history of exact sciences published in India can be cited. Lack of international perspective, and of awareness about current research work in the field is a general defect in the work of most Indians which is published here.

The article concludes with the following piece of advice (page 22) which is worth pondering even today, three decades after it was written, though arguably some things have changed during the interim:

Indians must give a serious thought to the point as to why, in spite of so much expenditure in the research and publication in the field, the situation is deplorable. But whatever be that, no one will disagree with a recent historian,³ that the history of Indian mathematics “still awaits a more reliable and scholarly treatment.”

In a similar vein, it may be worth mentioning here RCG’s response to the so-called “Vedic mathematics” of Bharati Krishna Tirthaji, which has captured popular imagination over the years for rather wrong reasons. In his article “In the Name of Vedic Mathematics” (gb-1988; II.2), he discusses “distinct categories of activities which have been going on in the name of `Vedic mathematics’” and analyses what can genuinely qualify to be called Vedic mathematics. He debunks various fantastic claims made about mathematical achievements supposedly found in the Vedas. In a section on Tirthaji’s system of “Vedic mathematics”, after recalling some of the basic facts about it and without undermining the mathematical contents of Tirthaji’s book, RCG notes (page 19) “…[W]e find that the title “Vedic mathematics” of the book or formulas have no more worth than that of fiction. An expert Sanskrit scholar even says that “to glorify the Vedas and Hindu culture by these false claims will only create revulsion of feeling when the truth is known” (Vishweshwaranand Indological Jour., Vol. IV, 1966, p. 109).”

Broad vision and an eye for detail is another trait of RCG

RCG’s commitment to the authentic pursuit of science, in spirit as well as in practical aspects, is manifest in his article “The Study of History of Mathematical Sciences in India” (gb-2001; II.4). The following quote, in a way, sums up his concerns about the study of the history of science in India (page 29):

Lastly, something about historical attitude and temperament. It is said that ancient India could not produce historians like Herodotus (note that epic and purāṇic tradition is mixed with myths). The reason is philosophical, i.e. more attention to spiritual matters. Indian scholars must cultivate greater historical sense. Also, we must have real love for records and try to preserve them. In foreign (western) countries, the papers, notes, correspondence of scientists is preserved and catalogued (in various libraries). In India material is often disposed off as waste paper …

It may be mentioned in this context that personally RCG is known to be very meticulous in record keeping.⁴ In an earlier article (gb-1990; III.4) on a related theme, after recounting various relevant details, he laments that in the case of India there is a serious problem concerning chronology with regard to prehistoric and ancient periods, with dates of most of the important events and works being subject to much controversy. One such controversy is the subject of the paper (gb-1987; II.1): On the date of Śrīdhara; thankfully the controversy seems to be resolved, and while we still do not know much details about him, it is settled that he is from the 8th century, and in particular preceded Mahāvīra (9th century) and not the other way as some historians had held.

Broad vision and an eye for detail is another trait of RCG’s work that is worth noting. There is an article by him with the intriguing title “Who Invented the Zero?” (gb-1995; III.6); a reader who may have a simple-minded answer to this like “the Indians” or “Āryabhaṭa”,⁵ will be induced by the article to think twice, or more. The author begins by noting “Obviously the answer depends on the meaning of `zero'”, and proceeds to peel the many layers that the idea of zero involves, providing a convenient categorization depending on the varied meanings, and to describe in that framework the developments towards the concept of zero in various ancient civilizations, from Egypt, Babylonia, Greece, China, and the Maya from South America. In the last section, bearing the title “Claims for Indian Zero” that aptly conveys the nature of inquiry taken up, the author considers various claims in the literature about it. He refutes some of them and discusses the merits and weaknesses of others, together with their chronology, putting them in perspective. No specific inferences are drawn regarding priorities, thereby bringing out the complexity of the question, and the multicultural heritage involved.

Attention to detail naturally goes hand in hand with an extensive body of references being cited with the papers. The material referred to by RCG readily stands testimony, apart from his oeuvre of writings, to his robust erudition. For someone with a vast knowledge, RCG is however rather restrained in offering opinion or analysis in various matters. It is not uncommon to encounter situations in the papers where one wishes the author would say more with regard to one or other peculiarity, but finds the text quietly moving on. Reticence sometimes seems to get an upper hand over other scholarly traits. This would, of course, provide the reader opportunities to explore possibilities and sort out one’s doubts with the help of the wealth of information received.

RCG’s work follows an approach driven by an inspiration to explore the truth as it may be

Incidentally, the title of the book, which translates as “joy of mathematics”, is taken from the pen-name⁶ that RCG used for many of his articles published in Gaṇita Bhāratī; indeed, over a dozen articles, including some short notes, were published by him under that name over the years. The choice of the name is an indicator of the joy RCG derives from the subject and that indeed is reflected in his work.

Overview of the Book

Let me begin this part with a description of the overall contents of the book. It features 46 papers of RCG, divided into eight parts, according to themes, including one each on bio-bibliographical writings on various historians of Indian mathematics (Part VIII), and on the theme of transmission of mathematics and astronomy from India to other civilizations (Part IX). The longest part, taking up over a quarter of the book, is on trigonometry (Part VII), an area to which some of the most important contributions of RCG belong. Parts III to VI contain writings on number systems, geometry and combinatorics; while contributions of the Jainas are involved in many papers, four papers involving specifically Jaina themes have been set out in a separate part (Part IV); incidentally one of the papers in this part is in Hindi.⁷ Papers on general historical topics relating to mathematics are in Part II, and some of the comments and quotations recalled earlier in the review are from there.

Of the 46 selected papers, 23 are from Gaṇita Bhāratī (an interesting simple proportion!) from the period 1983 to 2005, including five under the pen-name Gaṇitānanda mentioned above;⁸ (it may be recalled here that Gaṇita Bhāratī was started in 1979 with RCG as founding editor, and he nourished it in that role until 2005). Fourteen papers are from the Indian Journal of History of Mathematics (ijhs for short, in the sequel), a journal which has been a major vehicle for historical research in Indic sciences for over half a century since it was established in 1966. Of these 14 papers, seven are on his work in trigonometry published during 1967–1977, and the rest are from a later period, 1983–2013, except for one from 1973 on Jaina mathematics.

Among the papers published abroad there are three from Historia Mathematica (hm, in the sequel) which is associated with ICHM, one each from Centaurus (Cen), Historia Scientiarum (hs) and a volume Studies in the History of Exact Sciences – In Honour of David Pingree, edited by Charles Burnett, Jan P. Hogendijk, Kim Plofker and Michio Yano, and published in the series Islamic Philosophy, Theology & Science (Book 54), by Brill Academic Publishers, Leiden, 2004 (shes).

The paper in Hindi mentioned above is from Arhat Vacana, a journal specializing in Jaina studies. Another interesting paper, on magic squares, was published in the Bulletin of Kerala Mathematical Association (2005). One of the selected papers is also RCG’s Presidential Address at the 42nd Annual Conference of the Association of Mathematics Teachers of India (AMTI) in 2007, and appeared in Mathematics Teacher, India (mt), published by AMTI.

Apart from the papers of RCG, the book features, in the opening part, two articles on RCG: “A Birthday Tribute to R.C. Gupta” by Christoph J. Scriba⁹ which was published in Historia Mathematica, in the first issue of Volume 23, 1996, as a tribute on the occasion of RCG’s sixtieth birthday that had recently been celebrated, and “Professor R.C. Gupta Receives the Kenneth O. May Prize” by Kim Plofker,¹⁰ published in Gaṇita Bhāratī corresponding to the year 2009 (issued in 2011), in the context of his receiving the award.¹¹ Needless to say, both the articles exude considerable warmth towards RCG, and show a deep appreciation of his work. Incidentally, to the article of Scriba is appended a “Selected Bibliography of Radha Charan Gupta by Takao Hayashi”, with a list of 58 papers from the period until writing the article,¹² which is referred to by the author in the main article as “selected bibliography limited to some of the more extensive papers that Gupta has published in English”.

This description of the overall contents would be incomplete without the mention of certain other write-ups supplementing the material mentioned above. These include a Foreword by Roddam Narasimha¹³ conveying his deep admiration for RCG, a Preface with an elaborate discussion on the significance of RCG’s work in various respects, thus setting out the context and motivation for the volume, and also a write-up with the title “A Portrait of the Life of R.C. Gupta”, by the Editor, K. Ramasubramanian. It is seen that while the acquaintance of the latter with RCG is recent, he shares a warm relationship with RCG and we find in the write-up a fond rendering of the qualities of RCG, bearing a deeply personal touch, and references to some intense moments in their interaction. The book also contains a samarpanam (dedication) in the opening pages, and a Radhacharanaguptaprashastih (prashastih = praise) at the end, with eulogies to RCG in poetic form, in Sanskrit(!);¹⁴ English translations (prose) are provided.

Before concluding this overview, it would be worthwhile to say a few words on how a reader may be able to benefit from the book, in terms of getting to know the subject of ancient Indian mathematics closely. A dependable way of learning the subject would, of course, be through concerted and comprehensive effort over some years! Short of that a reasonable idea could be had from books containing full-fledged and reliable accounts (such as, for instance, those included in the bibliography at the end of this review). An issue with books, however, is that on account of their broad coverage, they usually do not find enough room for “small details” that make the topic of history interesting. On account of this, anyone with more than a “bookish” interest in the subject ought to also consult papers in the area, with good professional credentials. This need would be catered substantially by the work at hand, as it brings together many interesting and reliable papers scattered in the literature; the reader may be reminded here of my analogy with listening to tales from the past from a grandparent!

The Mathematical Papers: On Trigonometry

We shall now discuss the mathematical contents of the book. The aim here will be to put the contents in perspective, describing some background, and connecting up to the papers from the book in an overall fashion. In this respect, it would be convenient to begin with the topic of trigonometry to which RCG made some important contributions early in his career. When he entered the arena of the history of mathematics in the mid-1960s, a substantial corpus had been built up, in respect of work in arithmetic, algebra and geometry in ancient and medieval India, especially with the two classic volumes of Datta and Singh [6], published in 1935 and 1938 respectively (reprinted in 1962 as a two-part volume), and the Ph.D. Thesis (Ranchi University) of RCG’s teacher Sarasvati Amma which was later to materialise into the book [23], and there seems to have been a general sense that trigonometry is the major theme to be taken up next. Indeed, work had already begun. Datta and Singh had unpublished work, in Part III of their opus; the latter was subsequently edited and published by K.S. Shukla in the form of eight articles, on individual topics, in ijhs during 1980–1993.¹⁵ Thus it was that RCG set out to work on trigonometry in ancient and medieval India, at Ranchi University under the guidance of Sarasvati Amma, and completed his thesis on the topic in 1970/71. Regarding access to the work of Datta and Singh, RCG mentions in [10]: “I got one copy from S.N. Singh (son of A.N. Singh) and had referred to Part III in my doctoral thesis of which K.S. Shukla was one of the examiners.”

Papers in the compilation penetrate into the details to considerably deep levels

The pursuit of trigonometry in India goes back to the early centuries of ce. As is well-known, the notions of trigonometric functions in the present form, sine, cosine etc., originated in India, and even the name “sine” has its origin in the Sanskrit term jīvā for it, via a protracted route involving a long chain of transformations (see [5], for instance); the earlier Greek trigonometry, which inspired and influenced the pursuit of the topic in India during the early centuries of ce, was in terms of lengths of chords corresponding to the arc segments of the circle, in place of the half-chords involved in the definition of the sines. In India, the subject was known as jyotpatti-gaṇita.¹⁶ The functions involved were studied, not with respect to arcs of the unit circle as in modern trigonometry, but with respect to a circle of a chosen value for the radius (which was also the case with Greek trigonometry); the values generally varied with the work involved. Thus, jyā and koṭi-jyā associated with an angle \theta corresponded to R \sin \theta and R\cos \theta respectively, R being the chosen radius for the circle; in contemporary literature on the subject, it has been a practice to write \mathrm{Sin}\theta and \mathrm{Cos}\theta, with the initial letters in capital, for R\sin\theta and R\cos\theta, respectively, (and to refer to them as Sines and Cosines respectively) and we shall adopt the convention below; here R is subsumed in the notation, but this should cause no confusion. Sūryasiddhānta is perhaps the earliest extant work in India on the topic, but its period, believed to be around 300 ce, is uncertain (and the author of the work is also unknown). Among the reliably dated works, the Āryabhaṭīya (499 ce) of Āryabhaṭa (b. 476 ce) is the earliest in this respect, and the work has also been profoundly influential for the Siddhānta tradition of mathematical astronomy that flourished for around a thousand years, and also culminated in what is known as the Kerala school of mathematics (more on this later). Both these works provide, in particular, values of \mathrm{Sin}\theta for angles \theta that are multiples of \frac{1}{24}th of the right angle (=225', viz. 225 minutes), the radius being chosen to be 3438; this choice for the radius¹⁷ has to do with the fact that then the circumference is nearly 21600, which is the number of minutes corresponding to 360^\circ, so in particular for small angles \theta, \mathrm{Sin}\theta is numerically the same (approximately) as the angle in minutes.¹⁸ Āryabhaṭīya gives the 24 differences of the Sines of successive multiples of \frac{1}{24}th of the right angle, in a single verse¹⁹ and also describes two methods of computing them. Another notable classical astronomer-mathematician Brahmagupta also gives, in his Brāhmasphuṭasiddhānta (628 ce), values of the 24 Sines, but for radius 3270, and also a table for the successive differences of multiples of 15^\circ, for radius 150.²⁰

The paper (ijhs-1977; VII.9) contains a detailed discussion on the values of the radius R (the latter being referred to in the paper by the Latin term sinus totus for it) involved in various works, together with their background and motivation, and includes a table containing as many as 22 values (18, if one conflates those from later periods that are refinements involving fractional adjustments in the value 3438) and the dates of the listed works range over 550 ce to 1766 ce.

Various formulae involving Sines and Cosines (equivalently, sines and cosines) of angles were known from early times, and some were used in computations of the trigonometric tables mentioned earlier, but (equivalents of) the standard formulae for sines of sums and differences of angles are found only in the works of later authors, starting from Bhāskarācārya (b. 1114 ce). An interesting discussion concerning the latter formulae is given in (ijhs-1974; VII.7), where verses from original works for the formulae are quoted and interpreted, and several derivations of the formulae are described, along the lines of the works of various authors and commentators following Bhāskarācārya. Interestingly, one of the proofs is related to solutions of the so-called Pell equations, a topic which was dealt with in India since the time of Brahmagupta (7th century),²¹ and formed a strong point in the Siddhānta learning.

A rather striking fact about Indian trigonometry is that it was known from early on that the second-order differences of Sines, namely differences of the differences of successive multiples are proportional to the Sines; thus if S_n=\ \mathrm{Sin} n\theta, for an angle \theta, the \frac{1}{24}th part of the right angle for instance, and D_{n+1}=S_{n+1}-S_n then D_n -D_{n+1} is proportional to S_n.²² Āryabhaṭīya has a rule stating, in the notation as above, that D_{n+1}=D_n-S_n/S_1, and it is applied in one of the methods for computation of the table of the differences noted above. Apparently, the rule, and the application, was also known to the author of Sūryasiddhānta. The paper (ijhs-1972; VII.4) discusses the rule with reference to these works and also Golasāra and Tantrasaṅgraha of Nīlakanṭha, from 1500 ce.

Āryabhaṭa’s list of successive differences of the 24 Sines is in minutes in whole numbers, as 225, 224, …, which limits their accuracy. A more refined version giving a table of fractional correction terms, in seconds and thirds (60th and 3600th parts of a minute, respectively), is found in the commentary of Govindasvāmin (ca. 800–850) on Mahābhāskarīya of Bhāskara I (7th century). When adjusted by the correction terms that are provided (some of which are for being added to, while others to be subtracted from the original), the resulting values are seen to be remarkably accurate.²³ In (ijhs-1971; VII.3), which is considered one of RCG’s ground-breaking works, the table of correction terms and Govindasvāmin’s work towards determination of the finer values of the differences is discussed.

The Sines of angles were also often determined, without reference to the trigonometric tables, by an interesting formula giving the value of \mathrm{Sin} \theta, for an angle of \theta expressed in degrees, to be
\displaystyle \frac{R\theta (180 - \theta)}{\frac{1}{4}(40500 -\theta (180-\theta))}, where R is the radius of the circle involved; the original verse prescribes first computing P=\theta (180-\theta), and the value as above is given to be RP/\frac{1}{4}(40500-P). This formula is known after Bhāskara I, from the 7th century, in the light of its appearance in his Mahābhāskarīya; the formula also appears, in an equivalent form, in the Brāhmasphuṭasiddhānta (628 ce), but it is surmised that the former, while its date of composition is not known precisely, preceded the latter, and that Brahmagupta would have known the work of Bhāskara I, as a senior contemporary, though they belonged to rival schools.²⁴ Interestingly, the formula gives a very good approximation to the actual values, the error being less than 1\%, except for very small values of the angle²⁵ \theta is of the order of R\theta , while the above expression is of the order 4R\theta/225, so the percentage error is bound to be large.} and played an important role for centuries in the practice of mathematical astronomy in India. The paper (ijhs-1967; VII.1) traces this history, and presentations of the formula over a period in various Siddhānta works, including the Līlāvatī (1150 ce) and Grahalāghava (1530 ce). How the formula was arrived at is not known, and the paper discusses the possible routes, as many as five, that may have been adopted.

Pursuit of trigonometry continued robustly in the works of Bhāskarācārya (12th century) and Nārāyaṇa Panḍita (14th century). Subsequently, it got a major boost in the work of Mādhava, and the “Kerala school” consisting of a long teacher-student continuity (guru-śiṣya paramparā) that originated with him in the second half of the 14th century and flourished for well over two centuries. The work involved, in particular, the discovery of novel ideas and techniques that are basic to calculus, including (what are called) Gregory–Leibnitz series for the arctan function and Newton series for the sine function. While no mathematical composition of Mādhava has come down to us, works of many later stalwarts from the school, Nīlakanṭha, Jyeṣṭhadeva, and others, are known, in Sanskrit and Malayalam. Many of these classic works are now available also in English, edited and published in recent years, at long last.

Some of RCG’s papers have a link with this topic, and a glimpse of it may be had in Gaṇitānanda. The paper (ijhs-1969; VII.2) traces the history of interpolation formulae in the Indian context, related to the issue of finding values of sines of intermediate angles from the tabulated values, starting with Brahmagupta in the seventh century, covering the versions of Govindasvāmin (9th century) and Bhāskarācārya (12th century) and following it up with the formulae given by Mādhava and Parameśvara, from the 14th and 15th centuries respectively. He interprets²⁶ a verse of Mādhava (as recorded in Nīlakanṭha’s Āryabhaṭīya-bhāṣya and Tantrasa\dot{n}graha) as the formula \sin (x+\theta)=\sin x + \frac{\theta}{R} \cos x -\frac{\theta^2}{2R^2}\sin x, and the corresponding formula for cosine x, and notes that these are the initial terms of the Taylor series for the sine function. Parameśvara’s rules which also lead to the above formula are discussed.

In (hm-1974; VII.5), another of his well-appreciated papers, RCG describes a third order “Taylor series expansion” for the Sine function, from Siddhānta dīpika of Parameśvara. In this respect, three verses from the work are reproduced and translated. In the equivalent form, for \sin (x+\theta), the fourth term in this (following the expression recalled above) turns out to be -\frac{\theta^3}{4R^3}\cos x, in place of the correct expression -\frac{\theta^3}{6R^3}\cos x, as it should be in the Taylor expansion; rather disappointingly, RCG offers no discussion throwing light on the discrepancy,²⁷ and concludes only with “It is interesting that a four-term approximation formula for the Sine function so close to the Taylor series approximation was known in India more than two centuries before the Taylor expansion was discovered by Gregory about 1668 [Boyer 1968, 422]”. It may interest the reader to see the discussion in [21], with a geometric perspective on the formula.

A solution to a problem in spherical trigonometry, involved in astronomy, from Tantrasa\dot{n}graha is described in (ijhs-1974; VII.6); the problem concerns interrelation between the five quantities, terrestrial latitude, altitude of the Sun, declination of the Sun, the hour-angle, and the azimuth, and is dealt with by giving formulae for any two of them in terms of the other three. Answers to the ten cases that arise are described in the work individually. The relevant verses are translated and interpreted in the paper. The paper (gb-1987; VII.10), which is a survey of the achievements in medieval mathematics in South India, also includes some details on the topic as above.

The Mathematical Papers: On Numbers

We shall next discuss papers dealing, in a variety of ways, with numbers. As the discerning reader would recognize, the evolution of number systems is one of the crucial steps for the mathematical development of civilizations. It is also well-known that the decimal place value system for representing natural numbers,²⁸ in terms of the digits 1 to 9 and the zero, as it is now used around the world, attained its maturity in India and it was from here that it spread around the world. On the other hand, it does not seem to be well recognised that the progress from the early beginnings to the mature phase involved a long journey, and the details about its route are not very well understood.

In India people had a great fascination, in the Vedic as well as the Jaina and Buddhist traditions, for large numbers, especially the decuple terms, namely powers of 10 (that stood for “places” in their later avatar in number representation), which they chose to name up to quite large powers. This phenomenon was almost certainly instrumental, when writing of numbers set in, in 10 being chosen as the base for the place value system for representing numbers.²⁹ While the overall developments in this respect, starting from the names of decuple terms in the Vedas and other traditions has been known for long, and has been the subject of many books and other expositions, interesting new details have emerged sporadically when other previously unexplored manuscripts were studied. A standard scheme of place-names that prevailed over a significant part of history, described in particular in the works of Śrīdhara (c. 750 ce), Śrīpati (c. 1040 ce), Bhāskarācārya (1150 ce) and Nārāyaṇa Paṇḍita (1356 ce), gave names for powers of 10 upto 10^{17}, eka, daśa, śata, etc. going upto parārdha (the last term, incidentally, means half-way to `the other world’–-heaven).

In India people had a great fascination for large numbers

In Gaṇita sāra sa\dot{n}graha of Mahāvīra (9th century), there is a list of 24 consecutive decuple terms, upto 10^{23} (with substantial differences in nomenclature from what may be called the mainstream list). In the paper (gb-1983; III.1), after going over some of the history, RCG reported two extensions of Mahāvīra’s list, one from Mallana’s Telugu rendering of Gaṇita sāra sa\dot{n}graha, with 36 decuple terms, and another from Vyavahāra Gaṇita of Rājāditya, from the 12th century, in Kannada, with 40 terms (upto 10^{39}). In (gb-2001; III.7), where RCG revisited the topic, he notes also that while Mahāvīra and Rājāditya were Jaina, Mallana from the intervening period was a Śaiva (Hindu), and observes “Thus a sort of religious rivalry gave us bigger lists of decuple terms”;³⁰ it is also recalled in the end-notes to the article that al-Biruni had pointed out “religious reasons” involved also in the earlier list upto 10^{17}. In the same paper, RCG gives what he mentions as “the longest list of decuple terms that I have come to know recently”, from a work entitled Amalasiddhi, a Jaina canon, which has 97 consecutive decuple terms (upto 10^{96}). The reviewer found it striking that unlike other nomenclature schemes for decuple terms this one has the familiar current Hindi names hajāra, lākha, karoḍa, araba among them, at the appropriate place values! In another paper (gb-1990; III.3), the author describes a naming system from one of the editions of the Vālmīki Rāmāyaṇa, not with consecutive decuple terms, but advancing by a factor of 10^5, except, curiously, at one place (from 10^{47} to 10^{50}), starting with koti (10^7) and going upto 10^{60}. The scheme, termed the lakṣa scale, is used to describe the strength of the army of king Rāma! (Interestingly, notwithstanding the adoption of the lakṣa scale for the terms, the leading coefficient is seen to be koṭi.) The number describing the strength of the army then turns out to be of the order of 10^{67}, a rather huge number indeed;³¹[/latex].} the corresponding figures in older versions of the Rāmāyaṇa are less exorbitant, with one of them, for instance, giving a figure of the order of 10^{34}; the very critical Baroda edition in its main body (excluding footnotes dealing with variations from different editions) is noted to give the figure to be 1000 koṭi + 100 śanku (of the order of 10^{14}) and various versions are seen to have expanded on that!

Apart from large numbers, there was also a fascination for “infinity”. While such fascination extended to all the major traditions in India, it seems to have been more intense among the Jainas. An instance of this fascination may be found in the present volume in the paper (gb-1992; IV.3), “The first unenumerable number in Jaina mathematics.” The Jaina preoccupation with infinity went well beyond what is involved in the article, and in particular concerned various genuine types of infinity, but we shall not go into more details of this here.

In the development of the number system, especially in the written form, a crucial place is held by the notion of zero. While its use as a place holder served as a primary source for its sustenance in the initial period, at some stage its character as a number on its own was recognized, certainly by the time of Brahmagupta (7th century ce), who explicitly discussed arithmetic with zero, treating it as a number in its own right. As recalled earlier in this review, RCG has an interesting paper (gb-1995; III.6) on the topic.

As may be expected, aside from the exploratory and recreational engagement with numbers there is also a practical side reflected in ancient mathematical literature. While much of the economy was based on barter, there was nevertheless a significant mercantile community that dealt with money, in the form of gold coins etc., and in particular borrowings and interest on them etc. were involved; expositions of calculations concerning them are found in many ancient and medieval works including Āryabhaṭīya and Līlāvatī. Though this does not find much expression in Gaṇitānanda, there is indeed a short paper (gb-1993; III.5), “A problem of interest in Nārada purāṇa“, representing the theme; a verse from the said purāṇa which apparently had not been well understood previously has been interpreted by RCG, and it turns out to be about a comparison of two borrowing schemes with different interest rates!³² RCG notes that simple interest problems involving equalization, which the problem in question exemplifies, were very popular in ancient Indian mathematics and refers to his paper [9] (not included in the present compendium) for details.

Another important area involving numbers that held sway in ancient and medieval Indian works concerns combinatorial mathematics. Permutations and combinations, and magic squares are among the topics that were treated in substantial detail in Jaina as well as Hindu traditions since early times. These had religious, recreational as well as certain kinds of practical significance in their lives. There are three papers in the book under review, representing the general area. The paper (gb-1992; VI.1) introduces a method for computation of ^nC_r, successively in terms of the corresponding numbers for lower values of r (for given n), known as loṣṭa prastāra, described in the Bṛhat saṃhitā of Varāhamihira; it may be worth quoting here RCG’s comment in the paper “When the ancient Indian loṣṭa prastāra is depicted in this way, the so-called Pascal’s Triangle is at once seen formed in it.” Another paper in the area, (gb-1996; VI.2) is on a problem from Līlāvatī on the number of numbers with n digits in their decimal expansion for which the digital sum (sum total of the digits in the representation) is a given number S. The third paper alluded to, published in the Bulletin of the Kerala Mathematical Association (2005; VI.3), is on the topic of magic squares from an early period.³³ The idea of magic squares is claimed to be traceable in seed form to the Ṛgveda, based on a study by Christopher Minkowski, of a commentary of Nīlakaṇṭha Caturdhara composed in 1680 (see page 228). The author also describes in the paper the methods for construction of magic squares from Bṛhat saṃhita of Varāhamihira and a work called kakṣapuṭa by a Tantric Nāgārjuna, believed to be from sometime between the 7th and the 10th centuries. It may be mentioned here that the epilogue to the paper includes some interesting archaeological information about inscriptions containing magic squares.

The Mathematical Papers: On Geometry

Geometry has been a subject of much study in India since the ancient times, in the Śulvasūtras motivated by ritual purposes, in the Jaina canons in the context of their cosmological models, in the Siddhānta works in conjunction with mathematical astronomy, as well as in some miscellaneous works. An approach focussing on purely theoretical aspects has been rare however, and of later vintage. The reader is referred to the book of Sarasvati Amma [23] for a nice account of the subject. In the compendium under review, there are a few papers, which in fact touch some of the core topics, that we now discuss.

Next only to the rectilinear figures like squares, rectangles, triangles, with which we shall not concern ourselves here, given the scope of the set of papers under consideration, the circle is a figure that engaged the lives of ancient people, leading them to raise, and answer in ways that they could, various questions about it: How much is the area of a circle? How much is its circumference? In modern parlance, this amounts to understanding the value of the number \pi which is the ratio of the circumference to the diameter as well as of the area to the square of the radius.³⁴ At some point, methods for estimating \pi by realizing the circle as a “limit” (in a heuristic sense) of regular polygons with larger and larger number of sides, chosen to be double each time after starting with six, and using the Pythagoras theorem in conjunction to compute the perimeters, came into use in various cultures, whether independently or through transmission or interaction. In India, the value given in Āryabhaṭīya (499 ce) is perhaps the first in this category, though the Pythagoras theorem itself had been known for over a thousand years by then, as evidenced by the Śulvasūtras. Before this happened, people relied on values that they arrived at through speculation, coupled with intuition, perhaps guided by some practical experience with geometrical figures. In the Indian context these values primarily come from the Śulvasūtras and the Jaina canons. The Jainas are seen to have adopted the value \sqrt {10} for \pi, at least since the early centuries of ce. On account of its origin, it is often referred to as the Jaina value, but it was used also in the Siddhānta works and remained a popular choice in calculations even when more accurate values for \pi had became available. The paper (ijhs-1973; IV.1) gives an illustration of how the circumference of Jambūdvīpa, whose diameter is supposed to be 100,000 yojanas, was calculated using the value \sqrt {10} for \pi, namely as the square root of 10^{11}; the main focus of the paper, however, is on the specific procedure adopted for calculating the square root. There have been suggested explanations in literature about the choice of the value \sqrt {10} for \pi; one such explanation, going back to Mādhavacandra, a Jaina mathematician from the 10th century, and elaborations on that by various scholars (the original version being unclear) are discussed in (ijhs-1986; IV.2) by RCG, adding his own suggestions.

There is also a paper (Cen-1988; III.2), “New Indian values of \pi from the Mānava śulba-sūtra”, where RCG, after giving some background on the issue involved, has interpreted one of the verses from the Mānava śulvasūtra as giving a value of \pi to be 25/8. The first part contains some useful information, but about the proposed new value the reviewer would like to mention the following. While writing on the Śulvasūtras, in [1], the reviewer gave an interpretation of that verse; at that time the reviewer was not aware of RCG’s paper in question. The interpretation consists of a different geometrical construction, for describing a circle with the same area as a given square, that is akin to a construction in the Baudhāyana śulvasūtra for the same purpose; it yields a value of \pi that is about 3.1583. On the other hand, the reviewer has contended in [2], and also recently in [3] in some more detail, to the effect that the argument given by RCG for the new value proposed by him is not satisfactory. It would, however, be out of place to go into the details on this here.

The volume of a sphere is also seen to have been a source of much curiosity, going back all the way to the time of the ancient Jaina scholars. The rule given in Āryabhatīya, and adopted uncritically in many Siddhānta works turns out to be wrong; the reader is referred to RCG’s paper [13] for an interesting discussion on this. Bhāskarācārya, in the 12th century, was the first one to give a satisfactory formula in this respect, in the Indian context. Śrīdhara, from the 8th century, gave a formula for the volume as \frac{1}{2}(1+\frac{1}{18}) times the cube of the diameter. There is a curious issue associated with this formula: did Śridhara have the correct idea of the volume being \pi/6 times the cube of the diameter, and in the absence of a symbolic notation for the ratio \pi, or for some practical reasons, substituted an approximation for \pi as 19/6? Though this is possible, in principle, there is no evidence to support it, especially as there are no details found on how he arrived at the formula.³⁵ for \pi, on account of the widely accepted nature of that value, even though numerically it would not have been a good approximation. Bhāskarācārya’s formula is also in terms of an approximate value of \pi but the analogous issue does not arise in his case, on account of some inputs on how the formula was obtained. An interesting account of this saga is given in (hs-1991; V.1).

Questions of area and lengths were also considered in the Jaina literature for figures cut out from a circle by a chord (the smaller part, contained in a semicircle), known as bow figure on account of the obvious similarity of the shape with a bow. In those pre-trigonometry days they had curious formulae interrelating the diameter of the circle, lengths of the chord and the arc, and the area of the bow figure, apparently based on some extrapolations. A discussion on these, and other related formulae including some from other ancient civilizations, may be found in the paper (shes-2004; V.3). In a similar spirit, the later Jaina authors Mahāvīra (9th century) and Ṭhakkur Pherū (14th century) gave formulae for the area of a portion of a sphere cut out by a plane, in terms of the perimeter of the section and the height of the cut out part; this is discussed in the paper in Hindi, in Arhata vacana (2014; IV.4), and in his paper [14] (not included in Gaṇitānanda).

One of the important early discoveries in geometry in India is Brahmagupta’s formula for the area of a cyclic³⁶ quadrilateral as \sqrt{(s-a)(s-b)(s-c)(s-d)}, where a,b,c and d are the sides of the quadrilateral and s is the semi-perimeter (half the sum of the sides). Interest in cyclic quadrilaterals is seen to have revived later in the work of Nārāyaṇa Paṇḍita (14th century), and in Kerala mathematics. The paper (hm-1977; VII.8) discusses a formula given by Parameśvara, of the Kerala school, from around 1430, for the circum-radius of a cyclic quadrilateral, as \frac{1}{4A} \sqrt {(ab+cd)(ac+bd)(ad+bc)}, with a,b,c and d the sides of the quadrilateral and A the area, and its proof given in Kriyākramakarī of Śankara Vāriar (16th century); the proof involves, in particular, Brahmagupta’s area formula as above.

While the importance attached to yajnas in the Vedic period waned with the passage of time, the core element of fire worship continued to be involved in various Hindu practices. Though there do not seem to be adequate inputs to evolve an overall picture on the phenomenon, some known works point to the engagement. One such work is Siddhānta tattva viveka of Kamalākara from the seventeenth century. The work describes, in particular, the construction of agni-kunḍas in a dozen different shapes, including padma (lotus), yoni (vagina), pentagon and heptagon. The paper (gb-1998; IV.2) discusses various details in this respect, focussing on the geometrical aspects. RCG points out that by replacing the traditional geometric work out of the shapes, Kamalākara provided a simplified method of construction through the introduction of guṇakas (scaling factors). While the geometry involved in this is elementary, it is instructive as an application of simple geometric principles.

The Transmission of Mathematical Knowledge

It is mind-boggling to consider the massive flow of knowledge, even when the means of communication were very meagre, thanks only to the indomitable human spirit

In our age of handy possibilities of instantaneous communication across the globe, and being used to them, it is mind-boggling to consider the massive flow of knowledge that is found to have taken place, even when the means of communication were very meagre, thanks only to the indomitable human spirit. Apart from migrations, sometimes on quite large scales, accumulated knowledge of various geographically limited communities got passed on to others, on various accounts, including trade, conquests motivated by a variety of objectives, people moving in search of livelihood, and so on. I shall not dwell more on this except to note that what applies to the spread of ideas in general applies in particular to mathematical ideas, and that contrary to what people often tend to imagine, it is a multiple-way traffic (see, for instance, [16], Part I). It needs to be studied as such, with due objectivity, transcending the sensitivities associated with such an inquiry. It should also be borne in mind that the subject is nowhere near being cut and dried, ready for delivery. To quote RCG (gb-2001; II.4, page 25)

It seems History of Mathematics is not a dead but dynamic subject. Before 1930, it was mostly just the glory of the Greeks. After that the picture changed by findings in Babylonian mathematics. Further dimensions have been added to the ancient period by the researches in the prehistoric and megalithic times and by theories of ritual origin of sciences. Now medieval period is being enriched by studies and publications of Arabic mathematics. However, it is better to wait rather than give final immature judgements in a hurry. Let us pool material; building may be erected later on.…

RCG has done more than his share of pooling material towards erecting the building, not only in terms of contributing to evolve a picture of the state of knowledge in India at various times as we reviewed in the preceding sections, but also highlighting how the knowledge spread and was shared with other civilizations over the historical periods. The part “Transmission of Mathematics and Astronomy between India and Other Civilizations” in Gaṇitānanda has six articles illustrating the latter, including a short one (ijhs-1987; IX.3) which would also serve as a summary of the topic. We shall now discuss the contents of the articles in some detail.

More than 160 Chinese pilgrims and scholars came to India during the 5th to 8th centuries

As we noted earlier, the spread of numbers from India has been a big draw in the topic, and is the subject of (ijhs-1983; IX.2), titled “Spread and triumph of Indian numerals”. The article traces comprehensively the history of the spread of Indian numerals and computation, starting with its likely appearance in southern Europe by the 5th century,³⁷ in China by the end of the 6th century, and its definite arrival in the 7th century in Syria in the west and Sumatra and Cambodia in the east. A variety of documentation relevant to the issue is recalled from various periods until the 15th and 16th centuries, mapping the progress. The Indian numerals got introduced amongst the Arabs in the 8th century (more on this below). It is noted that al-Khwārizmī’s work was the first to expound the system systematically, in the 9th century, but his treatise achieved the greatest success only when introduced in the west through Latin translation, in the 12th century. The article also discusses the obstacles at various stages to progress towards wider acceptance of the system, including general apathy, orthodoxy, as well as an argument that the system is vulnerable to forgery. By the 16th century, however, the system came into widespread use in Europe, overcoming the hurdles, and eventually received universal acceptance.

RCG has done more than his share of pooling material

A systematic introduction of Indian mathematical sciences to the Arab world began in 772/773 ce when an embassy from Sind visited the court of Caliph al Manṣūr (755–775), who was very receptive to the influx of knowledge. On the academic front, al-Khwārizmī played an important role. The developments in this respect are discussed in (gb-1980; IX.1) and the article in Mathematics Teacher (mt-2008; IX.6). Apart from his role in promoting Indian arithmetic, and his innovations in algebra, al-Khwārizmī is also renowned as an astronomer. His astronomical tables became famous in the Islamic world extending up to Spain, and later in Europe through Latin translations. In his work, al-Khwārizmī made extensive use of Zīj al-Sinhind, which was a translation into Arabic of a Sanskrit astronomical work,³⁸ commissioned by Caliph al-Manṣūr. Al-Khwārizmī’s astronomical tables were in turn redacted by al-Majrīṭī who flourished in 11th century Spain, leading to the penetration of Indian astronomy into Spain. Al-Majrīṭī had several disciples who were instrumental in spreading it widely in Europe. This history and the developments stemming from it are described in (gb-1980; IX.1). In (mt-2008; IX.6), as the author mentions at the outset, several works based on or translated from Sanskrit sources are noted. The article includes a detailed listing of ten titles of the work on Indian sciences by al-Biruni, who is introduced as “doubtlessly the most famous scientist of Medieval Islam”.³⁹

There are two papers, (gb-1989; IX.4) and (gb-2005; IX.5), devoted to the interaction with China. Here, Buddhism comes through as the chief medium of the transmission. It is recalled that with the spread of Buddhism, translations were made in the early centuries of ce, of Buddhist works, which also touched upon topics in astronomy and arithmetic; the works and their contents have been discussed in some detail in (gb-2005; IX.5), thus putting the activity in perspective.
The article (gb-1989; IX.4) also presents some arithmetic problems from Chiu Chang Suan Shu (“Nine Chapters on the Mathematical Art”, the present text of which is from first century ce) and Chang Chhiu-Chien Suan Ching (“Arithmetical Classic of Chang Chhiu-Chien”, from the second half of the 5th century) akin to certain problems found in some Indian works of later period, by Bhāskara I (7th c.), Śrīdhara (8th c.) and Mahāvīra (9th c.), and it is suggested that these must have been sourced from the former, through the contacts that had developed.

The Buddhist education system gave birth to large-scale monastic universities, such as the famous Nalanda University, where, apart from Buddhist and Vedic religious studies, various secular topics such as medicine and mathematics were studied. Apparently, more than 160 Chinese pilgrims and scholars came to India during 5th to 8th centuries, some of whom carried back loads of Pali and Sanskrit works, many of which were later translated into Chinese. The period of the Tang dynasty (618–907) was a glorious period in China, during which there was considerable scientific interaction between China and India. A detailed account of visitors, both ways, and activities is given in (gb-2005; IX.5). After 900 ce, there was a setback in the interaction, due to changed political conditions in both the countries, but some contact continued, which has also been briefly described in the article.

On People and Their Work

Between the development of mathematics and the subject of its history, there is the crucial link, the historian of mathematics. Somehow, not enough gets written about the individuals whose labour is crucial to building the edifice of the subject. In a general way, this applies to mainstream mathematicians and scientists as well, but in the case of the history of mathematics, the issue seems to be more acute. It is therefore laudable when someone spares the time and effort to put together relevant information about the individual practitioners, for wider dissemination. For the same reason, it is also very welcome that in the compendium at hand we get introduced to some of the stalwarts, who have contributed to our understanding of the history of mathematics from ancient and medieval India. The contents are limited on account of a general lack of information, but nevertheless, this collection would be a valuable asset for a student of the history of mathematics.

Of the eight people on whom there are articles in the book, five are Indians: Sudhākara Dvivedī, M. Rangacharya, Bibhutibhushan Datta, T.A. Sarasvati Amma and K.S. Shukla. Mahāmahopādhyāya Sudhākara Dvivedī (1855–1910)⁴⁰ was indeed one of the early Indian pioneers in the process of nurturing the subject of the history of Indian mathematics, at an early stage, after Paṇḍit Bāpudeva Śāstrī, whom he succeeded at Benaras (Varanasi). He brought out authentic editions of several ancient works, augmented with his own commentaries in lucid Sanskrit. He also produced a book, Gaṇaka Tara\dot{n}giṇi (in Sanskrit), with biographical notes on ancient mathematicians and their work, a first of its kind. On the other hand, he also put in keen efforts for students to be exposed to modern mathematics, by writing books on modern topics at an advanced level in Sanskrit, and later in Hindi when it caught on as a medium of formal education. RCG’s luminous piece (gb-1990; VIII.4) introducing him concludes with the optimism “[M]ay this brief sketch inspire others to take up the writing of a fuller and more authentic biography of this remarkable man”. Unfortunately, the reviewer is not aware of any progress on this.

M. Rangacharya (1861–1916) is well-known for bringing out an edition of Mahāvīra’s Gaṇitasārasa\dot{n}graha in English, in 1912, thereby bringing new life to the work. A short article in Gaṇitānanda (ijhs-2013; VIII.8), written on the occasion of the centenary of publication of the edition, recounts various details about the author, the extant manuscripts of Gaṇitasārasa\dot{n}graha, its earlier translations into various languages during the interim since its composition in the 9th century, later versions using Rangacharya’s translation, and also some mathematical points from the work.

Not enough gets written about the individuals whose labour is crucial to building the edifice of the subject

Bibhutibhushan Datta (1888–1958) who is renowned for his book with A.N. Singh, History of Hindu Mathematics (mentioned earlier in this review), is also the author of numerous papers (over 50 in about 15 years during which he was active in research in the area) addressing many key issues concerning ancient Indian mathematics and endeavouring to clarify many misunderstandings generated by western authors in various respects. A short article of RCG (hm-1980; VIII.1) sketches the life of this pious man who became a sanyāsi (ascetic) in 1938, after a wandering life for five years as a prelude to that. During the ascetic life he was known as Swami Vidyaranya; there is also a later article by RCG on Datta [10], referring to the latter by his ascetic name in the title, corresponding to a talk he gave at the annual conference of the Indian Mathematical Society in 1987 at Gorakhpur.

Sarasvati Amma (1918–2000) and Kripa Shankar Shukla (1918–2007) have been two major historians of mathematics to have flourished in independent India and were instrumental in shaping the course of development of the topic in the second half of the twentieth century. As mentioned earlier, the former was RCG’s thesis guide at Ranchi. The articles (ijhs-2003; VIII.7) and (gb-1998; VIII.6) consist of short life sketches of the two, respectively, the former an obituary note and the latter an eightieth birthday felicitation note. More details about the two may be found in the articles [8] and [26], brought out as centenary tributes by Gaṇita Bhāratī.

Census of Exact Sciences in Sanskrit (CESS) symbolizes both–-the labour of love and the love of knowledge

There are also three foreigners concerning whom there are short notes in the book: David Pingree, Abraham Seidenberg, and Clas-Olof Selenius. The article (gb-1982; VIII.2) is a review of the Census of Exact Sciences in Sanskrit (CESS) brought out by Pingree, which has been a colossal achievement and to quote RCG “[N]o serious work on Indian exact sciences can be done without the CESS which symbolizes both–-the labour of love and love of knowledge”. It may also be recalled here that RCG wrote an obituary note [11] in Gaṇita Bhāratī, in the last volume that he edited.

The homage to Seidenberg is an obituary note (gb-1989; VIII.3) in which RCG observes that “Prof. Seidenberg has given us not only new findings but also some great ideas as applied to history of ancient mathematics” and cites, in particular, the hypothesis of diffusion and of ritual origin of mathematics.

Selenius is well-known for his deep study and expertise on the Cakravāla method for solving the so-called Pell equation Nx^2+1=y^2, in integers x,y for given N which is not a square. In (gb-1992; VIII.5), an obituary note, RCG provides a life-sketch and describes some of his work, noting in particular “[H]is exposition of its [cakravāla method’s] secrets is far illuminating than given by previous scholars, both Indian as well as foreign”. A letter received by RCG from Selenius, dated 7th February 1973, in response to one from him, is reproduced in the paper; the letter is highly illustrative of the scholar’s deep attachment to ancient Indian mathematics.

In Place of Conclusion

Gaṇitānanda thus gives a good exposure to a wide spectrum of topics on the history of ancient and medieval Indian mathematics. That includes mathematical topics, transmission and exchange of mathematical knowledge and also, to an extent, details on historians of mathematics. Viewed in totality, they can give the reader a fair idea of the subject. Being a compilation of well-researched papers, it penetrates into the details to considerably deep levels in the varied topics explored, and the resulting information content is unique. RCG expresses many wishes along the way towards strengthening the area and reforming the concept and practice of it, and I conclude this review with the hope that the book will contribute towards fulfilling them in the times to come. \blacksquare

References

• [1] S.G. Dani. “Geometry in the Śulvasūtras”, in Studies in History of Mathematics, Proceedings of Chennai Seminar, Ed. C.S. Seshadri. Hindustan Book Agency, New Delhi, 2010
• [2] S.G. Dani. “Cognition of the Circle in Ancient India”. Mathematics in Higher Education (in Russian: translation by Galina Sinkevich). 2016. 14: 61–76; available, in English, at arXiV: 1703.09645
• [3] S.G. Dani. “Some Constructions in the Mānava śulvasūtra”. Preprint, available at arXiV: 1908.00440
• [4] Bibhutibhushan Datta. “The Jaina School of Mathematics”. Bull. Cal. Math. Soc. 1929. 21: 115–145
• [5] Bibhutibhushan Datta and Avadhesh Narayan Singh, revised by K.S. Shukla. “Hindu Trigonometry”. Indian J. Hist. Sci. 1983. 18 (1): 39–108
• [6] Bibhutibhusan Datta and Avadhesh Narayan Singh. History of Hindu Mathematics: A Source Book, Parts I and II, Asia Publishing House, 1962; reprint: Bharatiya Kala Prakashan, Delhi, 2001
• [7] P.P. Divakaran. Mathematics in India: Concepts, Methods, Connections. Hindustan Book Agency and Springer Publications, 2019
• [8] P.P. Divakaran. “T.A. Saraswati Amma”. Gaṇita Bhāratī. 2018. 40 (1): 1–16
• [9] R.C. Gupta. “Some Equalization Problems from the Bakhśālī Manuscript”. Indian J. Hist. Sci. 1986. 21 (1): 51–61
• [10] R.C. Gupta. “Swami Vidyaranya and the History and Historiography of Mathematics in India”. Math. Student. 1987. 55 (2–4): 117–122
• [11] R.C. Gupta. “Prominent Polymath Pingree Passes Away”. Gaṇita Bhāratī. 2005. 27: 129–130
• [12] R.C. Gupta. “Śulvasūtras: Earliest Studies and a Newly Published Manual”. Indian J. Hist. Sci. 2006. 41: 317–320
• [13] R.C. Gupta. “Techniques of Ancient Empirical Mathematics”. Indian J. Hist. Sci. 2010. 45(1): 63–100
• [14] R.C. Gupta. “Mahāvīra-Pherū Formula for the Surface of a Sphere and Some Other Empirical Rules”. Indian J. Hist. Sci. 2011. 46(4): 639–657
• [15] Takao Hayashi. “Current Bibliography of Radha Charan Gupta”. Gaṇita Bhāratī. 2011 (issued in May 2013). 33: 137–172
• [16] George Gheverghese Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, 3rd Edition, Princeton University Press, Princeton, NJ, USA, 2011
• [17] George Gheverghese Joseph. A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact. Sage Publications, Los Angeles, CA, USA, 2009
• [18] S. Kichenassamy. “Brahmagupta’s Derivation of the Area of a Cyclic Quadrilateral. Historia Mathematica. 2010. 37: 28–61
• [19] Athanase Papadopoulos. “On the Legacy of Ibn Al-Haytham, An Exposition based on the Work of Roshdi Rached”. Gaṇita Bhāratī. 2014. 36: 157–177
• [20] Kim Plofker. Mathematics in India: 500 bce–1800 ce. Princeton University Press, NJ, USA, 2008
• [21] Kim Plofker. “The “Error” in the Indian “Taylor Series Approximation” to the Sine”. Historia Mathematica. 2011. 28: 283–295
• [22] K. Ramasubramanian and M.D. Srinivas. Development of Calculus in India, Studies in the History of Indian Mathematics, 201–286, Cult. Hist. Math., 5, Hindustan Book Agency, New Delhi, 2010
• [23] T.A. Sarasvati Amma. Geometry in Ancient and Medieval India. Motilal Banarsidass, Delhi, 1979
• [24] E.C. Sachau (transl.) Alberuni’s India. Reprinted, New Delhi, 1964
• [25] Raja Sridharan and M.D. Srinivas. “Folding method of Nārāyaṇa Paṇḍita for the Construction of Samagarbha and Viṣamagarbha Magic Squares”. Indian J. Hist. of Sci. 2012. 47(4): 589–605
• [26] M.D. Srinivas. “The Seminal Contribution of K.S. Shukla to Our Understanding of Indian Astronomy and Mathematics”. Gaṇita Bhāratī. 2018. 40: 17–51

Footnotes

1. As reported in his write-up “A Portrait of the Life of R.C. Gupta” by the editor K. Ramasubramanian, included in the book. ↩
2. Here and in the sequel gb stands for Gaṇita Bhāratī, followed by the corresponding year, and then the Chapter and section coordinates of the paper in the volume; similar abbreviations, in the same format will be used below for other journals in respect of papers referred to from Gaṇitānanda. ↩
3. There is a footnote mark in the original at this point, which refers to the book Introduction to History of Mathematics by Howard Eves, New York, 1969; page 182, with an addendum by RCG “Quotation is valid for 1989 also.”. ↩
4. Some interesting details on this can be found in “A Portrait of the Life of R.C. Gupta” by the Editor of the book under review, mentioned also in an earlier footnote. ↩
5. There is a prevalent notion among many people, including some professional scholars who are otherwise knowledgeable in history, that Āryabhaṭa invented the zero. How this baffling idea may have originated is not clear, but presumably, it is based on some `popular exposition’ by someone not adequately aware of pre-Āryabhaṭan mathematics, whether from India or elsewhere. RCG, in his article, only mentions, as far as Āryabhaṭa is concerned, that rules in the Āryabhaṭīya are based on place value system with zero (emphasis added for focussing). ↩
6. Actually the name as he wrote it was without the last “a”, a common north Indian feature in this respect, while for the book the version following the current conventions for spelling Sanskrit terms is adopted. ↩
7. It may interest readers who are not comfortable with reading papers in Hindi, to know that RCG has also an article on broadly the same topic in English [14], not included in Gaṇitānanda, where a good deal of the basic information from this paper can be found, though the two papers differ substantially in their overall coverage. ↩
8. This separate author identity of these articles has not been retained in the book. In this context it may be worth noting here that these are the papers with part and section coordinates II.1, II.2, II.3, III.3 and VIII.3. ↩
9. Christoph Scriba (1929–2013) was a renowned German historian of mathematics, affiliated to the University of Hamburg. He was awarded the Kenneth O. May prize in 1993, together with Hans Wussing. ↩
10. Kim Plofker, Union College, Schenectady, NY, USA, is a renowned expert and author of a well-received book and many papers on ancient Indian mathematics. She was a plenary speaker at the International Congress of Mathematicians 2010, held at Hyderabad, India. ↩
11. The prize was awarded in 2009, along with Ivor Grattan-Guinness, and was to be presented to him at the 23rd International Congress of History of Science and Technology, at Budapest that year. However RCG could not attend the Congress to receive it, and the award was subsequently ceremonially presented to him, at the valedictory function of the International Congress of Mathematicians, 2010, at Hyderabad, India, handed over by Kim Plofker. A photograph of RCG wearing the medal is included opposite to the contents page of the compendium. ↩
12. The selection in the book under review is restricted to papers on mathematics from India. Some of the papers in the list following Scriba’s article concern other civilizations, but there are also papers on ancient India that are not found here. On the whole, however, the present selection is adequately representative of the highlights of the work as far as Indian mathematics is concerned. A more comprehensive list of RCG’s papers, as of May 2012, may be found in [15]. ↩
13. For an introduction to Roddam Narasimha, the reader is referred to Bhāvanā, 1(2), April 2017 and 2(2), April 2018. ↩
14. These are mentioned in the Acknowledgements to have been essentially composed by K. Mahesh. ↩
15. Part II was published in 1938, Singh passed away in 1954, and Datta passed away in 1958. Thus, clearly, there is a long gap between the genesis of the contents of Part III and their publication, which has some uncertain history; some details in this respect may be found in (hm-1980; VII.1) and [26]. ↩
16. The term jyotpatti is derived from jyā, for chord, and utpatti corresponding to “output”; occurrence, as parts, of jyot and patti which might seem to have a familiar ring is coincidental!. ↩
17. The choice is shared by numerous other works from India as well. It is also speculated in the literature that this value of radius was adopted in the trigonometric tables (for chords) of the Greek astronomer Hipparchus from the second century bce, but there is not enough evidence to that effect, the tables themselves not being extant. Making the particular choice, with the motivation as indicated, involves using a value of \pi which is accurate to a certain degree, and while it can be seen that the approximate value for \pi given by Āryabhaṭa fits the choice, the same is not clear in the case of Hipparchus. ↩
18. The advantage involved in this may be compared with the usage of radians for measuring angles in modern trigonometry. ↩
19. It may be mentioned here, in passing, that this involves an alpha-numeric system of representing numbers that is peculiar to the work, and it was generally not used in later works. ↩
20. Smaller values of R are convenient for quick (mental?) computations, and though the results produced would be approximate, they would suffice for various practical purposes. ↩
21. An algorithm called Cakravāla for solving the equations was introduced by Jayadeva in the 11th century and popularized by Bhāskarācārya. ↩
22. This has been noted to be a finite form of the second- order differential equation satisfied by the sine function; the precise significance of such likeness is a matter of speculation. ↩
23. It may be mentioned here that in this table, the numbers are expressed in the Bhūtasa\dot{n}. ↩
24. In our times such an occurrence would come up as a case of plagiarism. Such an issue, however, is not relevant in their context. For one thing, the compositions were meant to be expositions, and not original contributions, though the latter did indeed appear in them, there having been no separate avenue, like journal papers today, for the dissemination of original ideas. Secondly, claims to the originality of statements apparently did not play much of a role, as it does today, presumably on account of the way academia were organised. There are many instances in ancient scientific literature which run counter to contemporary standards of “best practices”. ↩
25. It may be noted that for (sufficiently) small values of \theta, Sin \theta is of the order of R\theta , while the above expression is of the order 4R\theta/225, so the percentage error is bound to be large. ↩
26. Differing from an earlier interpretation of it by K.S. Shukla, that he rules out. ↩
27. Actually, as the formula is meant to have been derived, and not a product of inspired speculation, the cause of the discrepancy is something to wonder about. It may also be noted that for x=0 the result conflicts with the Mādhava–Newton series for the sine function, and the question arises as to how Parameśvara missed that. ↩
28. This should not be confused, or conflated, with decimal representation of real numbers in general (even for rational fractions), using the decimal point. That extension is a much later development, that took place in Islamic and western mathematics, bringing about much progress in arithmetic and in our understanding of the number world–-we shall however not go into any further details of this here. ↩
29. That this was not quite automatic is witnessed by the fact that, for a period, numbers were written in other styles as well. ↩
30. It may be observed that once a good understanding of the unending nature of the number system set in, the novelty of naming the decuple terms beyond those of practical relevance ought to have ebbed; for instance, we do not find the need for too many names for place-values now, it being very rare to use (non-composite) decuple terms beyond ko\d ti (10^7), or araba (10^9) in the Hindi-speaking part, in India, and trillion (10^{12}) worldwide. Apparently, the trend continued on account of competition in one or other context. ↩
31. It may be interesting to note here that on the Oxford University Press blog by Steve Cavill (https://educationblog.oup.com/secondary/maths/numbers-of-atoms-in-the-universe), the number of atoms in our solar system is estimated to be 1.2 \times 10^{56}. ↩
32. The reader is cautioned however that the Nārada purāṇa involved, while of uncertain date, is not very ancient, and could even be from as late as the 17th century. ↩
33. The topic of magic squares is treated in detail in Gaṇita kaumudi (1356 ce) of Nārāyaṇa Paṇḍita which, though mentioned, does not find adequate expression in the paper. A reader interested in the topic is referred to [25] and other sources traceable from there. ↩
34. Though we shall not dwell more on this point here, it may be noted that the equality of the two ratios involved is not automatic or evident from the perspective of uninitiated people, and only after some progress in mathematics that it came to be realized!. ↩
35. Such an inference would also have been justifiable, alternatively, if he had described the formula substituting \sqrt {10} for \pi, on account of the widely accepted nature of that value, even though numerically it would not have been a good approximation. Bh\={a}skar\={a}c\={a}rya’s formula is also in terms of an approximate value of \pi but the analogous issue does not arise in his case, on account of some inputs on how the formula was obtained. ↩
36. That Brahmagupta meant it to be only for cyclic quadrilaterals was missed by most Indian mathematicians down the line, and also by modern authors until recently. Brahmagupta is seen to have applied the formula only in that case, and it has been argued in [18] that the (unusual) term he used in stating the rule, tricaturbhuja, was in fact meant to be a technical term for a cyclic quadrilateral. ↩
37. The likely appearance in 5th century Europe, as alluded to, has however been disputed in the review of the paper in Mathematical Reviews, by M. Folkerts, asserting that the so-called “Geometry” of Boethius, on which the assertion is premised, is a forgery dating from the first half of the 11th century, and that there is no evidence that the “Gubar numerals” were known in the West before the end of the 10th century. ↩
38. The work in question is believed to be Brahmagupta’s Brāhmasphuṭasiddhānta; while RCG’s older article (gb-1980; IX.1) is silent on this, it is mentioned in (mt-2008; IX.6), where RCG notes (page 489) “This specification is supported by several ancient (e.g. al-Biruni) as well as modern scholars”. It is unclear however whether the assessment is based on actual comparison of contents of Arabic texts with the original Sanskrit text. The reviewer has also come across vague speculation that rather than Brāhmaspuṭasiddhānta, which is difficult and voluminous, the work involved may be Khanḍkhādyaka, a later work of Brahmagupta which is simpler. It may be recalled here that al-Biruni is noted to have brought out a corrected and revised Arabic translation of Khanḍkhādyaka, but with regard to Brāhmaspuṭasiddhānta, while Sachau in [24] (page 303) states that he was translating it, RCG comments (on page 492) “It is doubtful whether he could complete this tough task”. ↩
39. It seems to the reviewer that while the contribution of al-Biruni is undoubtedly very important from a historical perspective, especially in bringing out the role of India, from the point of view of scientific creativity, Ibn al Haytham, known also as Alhazen, wins hands down (see [19], for an introduction to Alhazen). ↩
40. The title was awarded to him by the then British Government of India for his distinguished scholarship and educational service. ↩