History of Science in South Asia
A journal for the history of all forms of scientific thought and action, ancient and modern, in all regions of South Asia
B O O K R E V I E W
MLA style citation form: Clemency Montelle, Book Review. History of Science in South Asia 2 (2014): r1–r6.
Online version available at: http://hssa.sayahna.org/.
HISTORY OF SCIENCE IN SOUTH ASIA
A journal for the history of all forms of scientific thought and action, ancient and modern, in all
regions of South Asia, published online at http://hssa.sayahna.org
Editorial Board:
1. Dominik Wujastyk, University of Vienna, Vienna, Austria
2. Kim Plofker, Union College, Schenectady, United States
3. Dhruv Raina, Jawaharlal Nehru University, New Delhi, India
4. Sreeramula Rajeswara Sarma, formerly Aligarh Muslim University, Düsseldorf, Germany
5. Fabrizio Speziale, Université Sorbonne Nouvelle – CNRS, Paris, France
6. Michio Yano, Kyoto Sangyo University, Kyoto, Japan
Principal Contact:
Dominik Wujastyk, Editor, University of Vienna
Email: ⟨hssa@sayahna.org⟩
Mailing Address:
Krishna GS, Editorial Support, History of Science in South Asia
Sayahna, jwra34, Jagathy, Trivandrum 695014, Kerala, India
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B O O K R E V I E W
Dr Sita Sundar Ram, Bījapallava of Kṛṣṇa Daivajña: Algebra in Sixteenth Century
India. A Critical Study (Chennai: Kuppuswami Sastri Research Institute, 2012)
287 pp. Rs 400/–. ISBNs 9788185170503, 8185170509.
The Bījapallava of Kṛṣṇa Daivajña is a substantial commentary on Bhāskara II’s
(fl. 1114) Bījagaṇita, composed some time in the late sixteenth century. Kṛṣṇa
Daivajña came from an impressive lineage of famous jyotiṣīs and was an active scholar at the Mughal court in Agra. The author of an original work on
eclipse reckoning, the Chādakanirṇaya, and several other commentaries, including Bhāskara’s Līlāvatī and Śrīpati’s Jātakapaddhati, Kṛṣṇa Daivajña was also part
of a group of scholars at the court who were responsible for translating Ulugh
Beg’s astronomical tables into Sanskrit. He is therefore a key figure in secondmillennium astral sciences in India. Dr Sita Sundar Ram’s expository study on
the Bījapallava is a welcome addition to the field as it makes more accessible the
writings of this important scholar.
Commentaries are valuable for a number of reasons. Most directly, they are
crucial for helping the reader understand the text they are analysing. In many
instances, they can clear up confusion caused by often tersely expressed rules
or concepts in the base text by parsing the grammar, paraphrasing expressions,
giving synonyms to technical terms, providing a worked example, or sometimes
even deriving the parameters invoked by the original author. But more broadly,
commentaries give modern scholars a sense of the status of the text in the
time contemporary with the commentator. They can also offer vital insight
into deeper issues relating to the ways in which the technical content was
generated, justified, and understood by those practitioners who used the texts.
Modern scholars have deemed this latter point to be particularly apt when it
comes to the Bījapallava. Kṛṣṇa Daivajña provided a commentary to Bhāskara’s
text which included demonstrations or derivations of the mathematical rules
and relations (the so-called upapatti), mock debates between himself and an
imagined interlocutor, analyses of the mathematical concepts that underpin the
algorithms, such as zero, unknown quantities, negative numbers, explorations
history of science in south asia 2 (2014) r1–r6
2 Book Review
on issues regarding determinability and conditions for solvability, and the like.
This work is therefore a rich and nuanced source for our understanding of the
practice of mathematics in the Indian subcontinent.
Kṛṣṇa Daivajña’s Bījapallava has been edited several times, with editions by
Āpaṭe et al. (1930), by Radhakrishna Sastri (1958), and by Vasistha (1982). Dr
Sita Sundar Ram bases her study on these three editions and presents excerpts
of key passages which she translates and analyses with detailed mathematical
worked examples. She also includes a variety of cross-references to related
mathematical discoveries throughout her analysis where appropriate. These are
drawn from other works of Bhāskara, from related authors in the same tradition,
and also from a variety of Eurasian cultures of inquiry. She also offers modern
insightful reflection on the scope and context of the mathematical features that
appear throughout the work.
The book opens with a brief survey of Indian mathematics, followed by five
chapters each dedicated to a section from the Bījapallava. These are: six mathematical operations, the pulverizer (kuṭṭaka), indeterminate equations of the
second degree, equations with one unknown, and lastly equations with many
unknowns. The work is then wrapped up with an appraisal of Kṛṣṇa Daivajña’s
expertise in other areas, including philosophy, grammar, prosody. An account
of his reception by later authors considers his scholarly legacy and several
appendices contain useful reference information including the 219 verses of
Bhāskara’s Bījagaṇita and an ample glossary of technical terms.
The Bījapallava is a demanding work and many of the mathematical problems
discussed therein are complex. Mathematical highlights that Dr Sita Sundar
Ram brings to the reader’s attention include: Kṛṣṇa Daivajña’s characterisation
of moving east or west on a number line (p. 19) to conceptualise the effects of
operating with positive and negative numbers; the discussion on the ‘symbols’
for representing unknowns and their implication (p. 30ff); analysis of the pulverizer (kuṭṭaka method, p. 39ff); the classification of equations with one or more
unknowns (p. 119–122); commentarial confusion over the lack of integer solutions for a particular indeterminate equation of the second degree (p. 105–107),
to name a few.
Dr Sita Sundar Ram notes on several occasions that a distinctive feature of
Kṛṣṇa Daivajña’s commentary is his inclusion of upapattis, often translated as
“proof” or “demonstration” or “derivation”. She notes (p. xiv) upapatti can
have two senses: it is used “in the sense of both proof and method”. This
particular topic is becoming increasingly important in modern scholarship
when considering the salient features of mathematical practice in India as
it gives historians a window into some of the ways in which mathematical
history of science in south asia 2 (2014) r1–r6
Clemency Montelle 3
propositions were verified and how and why the original authors believed
their results to be justified, an area which has long been under-studied. Examining key textual passages amongst the original authors on this topic helps
us understand the role of demonstration in the astral sciences.1 Upapattis in
commentaries are also critical for better comprehending the many ways in
which commentators responded to the base texts and the strategies the deemed
useful and relevant for the interpretation of mathematical material (including
udāharaṇa, artham, vāsanā, utpatti, and the like) . The critical nature of these types
of commentarial passages and what they involve ultimately helps us understand
the role and function of the commentary and sheds light on some of the deeper
epistemological issues relating to the status of mathematical knowledge in this
tradition.
As more critical scholarly publications of commentaries become available, scholars are in a better position to reflect on the genre. Several recent studies have explored the broader role of the commentary and its constituent parts as an integral
part of scientific industry. Bronkhorst (2006), for instance, considers the ways in
which commentaries support the texts they comment on. He observes that while
the primary function of the commentary is to clarify the fundamental text, that
often in the process these technical exegeses go beyond this aim, and develop
and advance the content in ways that the original authors might never have anticipated. Dr Sita Sundar Ram provides many examples which support these more
general reflections. For instance, one of the examples she presents (pp. 212–215)
is a worked solution to a bhāvita problem (expressions dealing with the product
of two unknown variables). The problem amounts to solving 4x + 3y + 2 = xy
for integer x and y.
2
She notes that Bhāskara stated that such equations can be solved both ‘algebraically’ and ‘diagrammatically’, although he didn’t provide the latter.3 Kṛṣṇa
Daivajña’s commentary supplies a worked solution specifically using diagrams,
where the products of the constants and unknowns are imagined to be rectangles with yet to be determined sides and various diagrammatic manoeuvres
produce the unknown ‘lengths’. Her careful and methodical treatment of Kṛṣṇa
Daivajña’s account with accompanying diagrams and identification of the vari1See, for instance, the discussion in Sarma et al. (2009, pp. 267–310), In particular, Kṛṣṇa Daivajña’s upapatti of the kuṭṭaka process is translated and discussed in Appendix B.
2Bhāskara Bījagaṇita verse 204 (Sita Sundar Ram 2012: 267):
catustriguṇayo raśyoḥ saṃyutir dviyutā tayoḥ |
rāśighātena tulyā syāt tau rāśī vetsi ced vada ||
3Here, Dr Sita Sundar Ram translates kṣetra (lit. a figure) as ‘geometrically’, but what this
amounts to is using diagrams.
history of science in south asia 2 (2014) r1–r6
4 Book Review
ous steps of working with the resulting rectangles, gives a sound appreciation of
the original mathematical steps of working. A reproduction of a page from one
of the editions can be seen in figure 1. Of course, this edition may be quite different from the way in which the scribes presented the text in their manuscripts,
however it gives something of an impression of the layout and aspects invoked
when solving this problem.
Dr Sita Sundar Ram includes many upapattis throughout the work, sometimes
supplementing upapattis from other authors for completeness. Her decision to
paraphrase Kṛṣṇa Daivajña’s upapattis using modern mathematical terminology
has the benefit of making the mathematics tractable to the modern reader,
however her symbolic summaries which are expressed using modern algebraic
forms of reasoning can obscure the original phrasing and processes which are
key to gaining insight into the original thought processes and practices of the
Kṛṣṇa Daivajña. Readers are left with many questions. How precisely were
these original ‘equations’ expressed in prose? And how were they manipulated,
reduced, and simplified and how were various symmetries and similarities
spotted without symbolic styles of reasoning? What is the language and
grammar of these upapattis and how do they contrast with other technical
exegetical passages? All of these features are vital to understanding more fully
the contrasting and rich ways of operating in the specific circumstances that
commentators such as Kṛṣṇa Daivajña were working in. There is great potential
here to expand on this topic.
For example, a theme of considerable interest is Bhāskara’s treatment of interdeterminate equations of the second degree (invoked by the Sanskrit compound
vargaprakṛti) which can be captured in modern symbolic notation as Nx2 + 1 =
y
2
. Before turning to Bhāskara’s approach to solving expressions such as these,
Kṛṣṇa Daivajña presents the details of earlier mathematician Brahmagupta’s approach, known as the bhavana method. Kṛṣṇa Daivajña gives three upapattis
to illustrate this method. Dr Sita Sundar Ram begins her depiction of Kṛṣṇa
Daivajña’s commentary by setting up auxiliary equations Nx2
1 + k1 = y
2
1 and
Nx2
2 + k2 = y
2
2
, multiplying both sides by y
2
2 and so on, working through entirely algebraic steps of reasoning to derive the final expressions which generate
the required integer solutions. However, Kṛṣṇa Daivajña’s passage is entirely
in prose. It employs specific inflections of grammar and technical vocabulary
to express relations and operations between unknowns, which are themselves
expressed by lexical units jye for jyeṣṭha; kṣe for kṣepa and ka for kaniṣṭa. Kṛṣṇa
Daivajña’s ‘auxiliary equations’ are thus expressed: āka 1 ājye 1 ākṣe 1 dvika 1 dvijye
1 dvikṣe 1. Algebraic substitutions and simplifications, grouping of like terms, indexical relations, and the like do not ‘appear’ quite so obvious in prose, so the
process of reworking equations to new expressions is achieved quite differently.
history of science in south asia 2 (2014) r1–r6
Clemency Montelle 5
Figure 1: A page from Āpaṭe et al.’s edition of the Bījapallava. Here, Kṛṣṇa Daivajña is detailing
the way to solve a bhāvita problem diagrammatically.
history of science in south asia 2 (2014) r1–r6
6 Book Review
While taking a dozen or so lines of modern symbolic reasoning, Kṛṣṇa Daivajña’s
explanation takes over three pages of printed commentary! (See, for instance
pp. 93–96 of Āpaṭe et al.’s edition). All of these features are highly pertinent for
understanding more fully the distinct mathematical practices in this culture of
inquiry. Indeed, while modern historiographical practise in recent decades has
tended to overly denigrate those approaches to original text which use the resources, symbolic and otherwise, of modern mathematics, the methodological
choice to depict the mathematical content of the text with modern symbolism
does have a place in historical expositions. However, it must be employed with
care.
Overall, preparing a technical text such as this presents many challenges. It
requires broad mathematical expertise and mastery of Sanskrit, as well as historical sympathy and reflection. Dr Sita Sundar Ram’s comprehensive study
of this seminal work is thus an impressive and commendable contribution and
her achievement has provided a solid foundation both for reference and further investigation. The work is well organised, comprehensive, and amply referenced. Sound mathematical analysis underscores the textual passages, clarity is
provided on some fairly complex and intricate mathematical passages, and her
work will be accessible to a wide audience including mathematicians and historians alike. This publication will serve as a important source for contributing to
our knowledge of second-millennium astral sciences in India and is an incentive
to continue to explore and study critically commentaries such as these.
REFERENCES
Āpaṭe, Dattātreya Viṣṇu et al., eds. (1930). Bhāskarīyabījagaṇitam. KṛṣṇadaivajñaviracitaNavāṅkuravyākhyāsahitam. Pune: Vināyaka Gaṇeśa Āpaṭe at the Ānandāśrama Press. url: http :
//tinyurl.com/bijapallavam1930.
Bronkhorst, Johannes (2006). “Commentaries and the History of Science in India”. In: Asiatische
Studien 60, pp. 773–88. url: https://www.academia.edu/3279438/.
Radhakrishna Sastri, T. V., ed. (1958). Bīja Pallavam (a Commentary on Bīja Ganita, the Algebra in
Sanskrit). Kṛṣṇagaṇakaviracitam Bījapallavam (Bījagaṇitavyākhyā). Tanjore: S. Gopalan for the T.
M. S. S. M. Library,
Sarma, K. V. et al. (2009). Gaṇita-Yukti-Bhāṣā (Rationales in Mathematical Astronomy) of Jyeṣṭhadeva:
Volume I: Mathematics Volume II: Astronomy. Sources and Studies in the History of Mathematics
and Physical Sciences. London and New Delhi: Springer and the Hindustan Book Agency.
Vasistha, Biharilal, ed. (1982). Bījagaṇita with Navāṅkura. Jammu: Ranavir Kendriya Samskrita
Vidyapitha.
Dr Clemency Montelle
University of Canterbury
New Zealand
history of science in south asia 2 (2014) r1–r6
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