For a time it was possible for historians of mathematics to believe, with Morris Kline1, that the notion of a “proof” played no role in the mathematics of classical times in South Asia, but the labors of both Asian and Western scholars in the twentieth century, most notably in the area of publishing and translating Indian commentarial texts, have clarified for all of us the both importance the Indians assigned to mathematical proof (in Sanskrit, yukti or upapatti) and the meticulous care with which careful rationales for mathematical results — from quite basic arithmetical procedures to what are nowadays understood as power-series representations of trigonometric functions — were constructed and communicated.
Insofar as they made no systematic effort to identify a set of basic principles from which all other results could be deduced, Indian mathematicians (ganakas, “reckoners”) did not work axiomatically. Instead the Indian approach to upapatti is informal and flexible, approximating closely to David Henderson’s student-friendly view of proof as “a convincing argument that answers a why-question.”2 Accordingly, Indian ganakas made free use of geometrical constructions to establish algebraic rules and were willing to develop a rationale using a particular case or particular numerical values so long as it would be clear to their readers how to formulate the argument in general. Indian upapatti had nothing to do with any program to reduce mathematics to logic; rather, proofs were offered in the context of the larger task of enhancing understanding on the part of the student.
The long-awaited publication of the Ganita-Yukti-Bhasa (GYB) by Jyesthadeva (circa 1550 CE) makes available to modern readers the mathematical argumentation that underlies the remarkable achievements of a school of astronomer-mathematicians that flourished in Kerala, South India from approximately 1400 to 1600 CE.3 The signature results of the Kerala School, generally attributed to Madhava Sangamagrama, include what are known in the West as the MacLaurin series for the sine and cosine functions, and the infinite series for π that was derived in Europe by Leibniz and Gregory more than two centuries after Madhava’s work.
Volume I of the GYB treats mathematics in general, beginning with simple arithmetical rules including efficient methods of multiplication, squaring and the extraction of square roots, arithmetical operations with fractions, and the principle of proportionality known in Indian mathematics as the Rule of Three. Jyesthadeva continues with the Kuttakara (the “pulverizer”), a method for solving linear indeterminate equations that is applied in Indian astronomy to problems about planetary motion.
The author next turns to results about the circle and its circumference, and it is here that we find the proof of the so-called Pythagoran theorem (known in India as bhuja-koti-karna, “arm-upright-diagonal”), as well as the derivation of the infinite series for π. More precisely, the Kerala School provides an infinite series for the circumference C of a circle in terms of its diameter d:
C = 4d (1 – 1/3 + 1/5 – …).
The rationales for series results in the Kerala School involve a form of infinitesimal analysis that calls to mind the early stages of calculus in Europe, and hence they have elicited much interest in the contemporary mathematical community. Less well-known but equally worthy of attention are the methods, developed by Madhava and discussed here by Jyesthadeva, for adding a “correction term” to any given partial sum of this slowly-converging infinite series, so as to yield impressively accurate approximations to the circumference itself.
The final portion of the GYB derives the power series representation of trigonometric functions, again employing infinitesimal analysis, continues with a series of exquisite theorems about cyclic quadrilaterals, and closes with miscellaneous results such as the volume and surface area of a sphere.
In addition to Sarma’s translation of the GYB, Ramasubramanian, Srinivas and Sriram provide extensive and invaluable explanatory notes on Jyesthadeva’s derivations, using modern mathematical terminology and symbols. A stimulating epilogue entitled “Proofs in Indian Mathematics” gives further examples of upapatti in Indian mathematical texts and sets forth the editors’ views regarding the “classical Indian understanding of the nature and validation of mathematical knowledge”, a subject that is likely to come in for much future study as the later commentarial texts continue to be edited and translated.
Volume I (Mathematics) will be of the greatest interest to the mathematical community, and is well worth inclusion in any library, even the libraries of institutions oriented towards undergraduate teaching. Along with other recent publications (see especially Kim Plofker’s contribution to The Mathematics of Egypt, Mesopotamia, China, India and Islam edited by Victor Katz) it functions as rich and accessible collection of primary-source material for undergraduate classes in the history of mathematics.
It should be kept in mind that, in classical India, applications of mathematics to astronomy and astrology motivated so much of the development of mathematics that the three subjects were generally treated as a unified discipline (jyotishastra, “star science”). Therefore Volume II (Astronomy) is recommended for any collection in the history of mathematics, especially since the editors’ critical notes constitute a valuable exposition of computational methods in Indian astronomy.
1 See, for example: “(With the Hindus) there is much good procedure and technical facility, but no evidence that they considered proof at all. They had rules, but apparently no logical scruples …” [Mathematical Thought from Ancient to Modern Times, Oxford 1972, p. 190]. Such views were probably based on a reading of root-texts in the Indian mathematical tradition, in which mathematical procedures were recorded as a series of terse formulations designed for easy memorization by the student. Proofs tended to show up in the commentaries, which received less attention from early European Indologists.
2 See Henderson’s Experiencing Geometry in Euclidean, Spherical and Hyperbolic Spaces, Second Edition, Prentice Hall, pp. xxix-xxxii.
3 Most major treatises in Indian mathematics were composed in Sanskrit, the common language of scholarship in classical times, but the GYB is written in Old Malayalam, a local vernacular that is unknown to most historians of Indian mathematics. A Sanskrit version of the GYB actually exists; it was edited by the late K.V. Sarma and published by the Indian Institute of Advanced Study in 2004. Sarma describes it, however, as “a rough and ready translation into Sanskrit if the Malayalam original by one who lacked not only the ability of writing idiomatic Sanskrit but also an adequate knowledge of the subject.” (See p. xl of the Introduction.) For this reason it is not helpful, even for most scholars of Indian mathematics, as a guide to the contents of the GYB itself.
Volume I: The Eight Mathematical Operations; The Ten Questions and Answers; Arithmetics of Fractions; Rule of Three; Kuttakara; Circle and Circumference; Derivation of Sines.
Volume II: Computation of True Planets; Earth and Celestial Spheres; The Fifteen Problems; Gnomonic Shadow; Eclipse; Vyatipata; Maudhya and Visibility Corrections of Planets.