Studies in the History of Indian Mathematics

David Edwin Pingree (2 January 1933–11 November 2005) employed the fifty years of his scholarly career investigating the development of mathematics, astronomy and the related exact sciences from ancient Mesopotamia to early modern Europe and India. He published editions, translations and studies of source texts in Akkadian cuneiform, Greek, Latin, Sanskrit, Arabic and Persian, on subjects ranging from infinite series and interpolation techniques to astral magic and iconography in astrological texts. He was professionally affiliated with Harvard University as an undergraduate (B.A. in Classics and Sanskrit, 1954), graduate student (Ph.D. in Sanskrit and Indian Studies, 1960), and Junior Fellow (to 1963); the University of Chicago as a faculty member in the Oriental Institute and Departments of History, South Asian Languages, and Near Eastern Languages (1963–1971); and Brown University as a professor in the Departments of Classics and the History of Mathematics (1971–2005). Over the course of his immensely productive career he received many honors, including a Fulbright Scholarship, a Guggenheim Fellowship, and membership in several learned societies, among them the American Academy of Arts and Sciences, the American Philosophical Society, and the Institute for Advanced Study at Princeton University. Pingree was awarded the title of “Abhinavavarāhamihira” by the government of Uttar Pradesh in 1979, and in 1981 was one of the first recipients of the MacArthur Fellowship (popularly nicknamed the “Genius Grant”), together with co-honorees including the philosopher Richard Rorty, the paleontologist Stephen Jay Gould, and the computer scientist Stephen Wolfram.

Born at Chengannur in Kerala on 27^th December 1919, Krishna Venkateswara Sarma had his school education in Attingal near Thiruvananthapuram. He completed his B.Sc. degree with Physics as the major subject in 1940, from Maharaja’s College of Science, Thiruvanathapuram. His family tradition of Sanskrit scholarship influenced Sarma to join the M.A. course in Sanskrit at Maharaja’s College of Arts, Thiruvanathapuram, which he completed with distinction in 1942. During 1943–51, he was in charge of the Manuscripts Section of the Kerala University Oriental Research Institute and Manuscripts Library. It is here that he acquired expertise in deciphering and critically editing palm-leaf and paper manuscripts of Sanskrit and Malayalam texts. During this period, he prepared an analytical catalogue of nearly 50,000 manuscripts of the library.

Yajnas, or fire rituals, formed an integral part of life in the Vedic culture, going back to 1500 BCE or earlier, and extending until about the sixth century BCE. Some of these concerned sacrifices to be performed regularly by a householder (gṛhastha), while performance of certain others was prescribed for bringing about fulfilment of specific aims or desires, which included both material (acquiring cows, vanquishing an enemy etc.) and transcendental (securing place in heaven) aspects.

IT IS WIDELY believed that zero originated in Indic Civilization but the evidence in support of that belief is not only meager; it is almost zero. No place or time, let alone the name of a discoverer or inventor, has ever been suggested. How can we handle such a problem? We must start from the beginning.

Six combinatorial tools (called pratyayas) have been in systematic use in India for the study of Sanskrit prosody (Chandas-śāstra) and these go back in time at least to Pingala (c. 300 BC?). Among these, three—prastāra (an enumeration rule for generating all the possible metrical patterns of a given class as a sequence of rows), uddiṣṭa (the process for finding, for any given metrical pattern, the corresponding row number in the prastāra) and naṣṭa (the converse of uddiṣṭa)—are found in Bharata’s Nāṭyaśāstra, in the chapter where prosody is discussed. Incidentally, the problem of placing Piṅgala and Bharata in the chronology of time still remains an unsettled question. The notion of pratyayas was perhaps discussed in other ancient texts of music also. However, the first extant text on music where the pratyayas are systematically dealt with, both in connection with patterns of musical phrases (tānas) and patterns of musical rhythms (tālas), is Saṅgītaratnākara of Śārṅgadeva (c.1225 AD). Nārāyaṇa Paṇḍita in his Gaṇitakaumudī (1356 AD) deals with some of these questions in a more general context, though his theory does not cover the kind of tāla-prastāra considered by Śārṅgadeva.

I was flabbergasted when I first read Augustus De Morgan’s writings about negative numbers¹. For example, in the Penny Cyclopedia of 1843, to which he contributed many articles, he wrote in the article Negative and Impossible Quantities:

It is not our intention to follow the earlier algebraists through their different uses of negative numbers. These creations of algebra retained their existence, in the face of the obvious deficiency of rational explanation which characterized every attempt at their theory.

In fact, he spent much of his life, first showing how equations with these meaningless negative numbers could be reworked so as to assert honest facts involving only positive numbers and, later, working slowly towards a definition of abstract rings and fields, the ideas which he felt were the only way to build a fully satisfactory theory of negative numbers.

Ancient Indian mathematical treatises contain ingenious methods for finding integer solutions of indeterminate (or Diophantine) equations. The three greatest landmarks in this area are the kuṭṭaka method of Āryabhaṭa for solving the linear indeterminate equation ay − bx c c, the bhāvanā law of Brahmagupta, and the cakravāla algorithm described by Jayadeva and Bhāskara II for solving the quadratic indeterminate equation Dx² + 1 = y². We shall briefly recall the history of the above equations in ancient India and of their rediscovery in Europe, give an account of the ancient Indian texts dealing with algebra in general and the above indeterminate equations in particular, and mention a few works on history of mathematics which have highlighted ancient Indian algebra, especially indeterminate equations. We shall then discuss various mathematical aspects of the kuṭṭaka, bhāvanā and cakravāla from the viewpoint of modern algebra and number theory and the general cultural atmosphere in which the leading Indian mathematicians undertook such explorations. We shall also examine the coverage of these results in texts involving history of mathematics.

In his pioneering history of calculus written sixty years ago, Carl Boyer was totally dismissive of the Indian contributions to the conceptual development of the subject.¹ Boyer’s historical overview was written around the same time when (i) Ramavarma Maru Thampuran and Akhileswarayyar brought out the first edition of the Mathematics part of the seminal text Gaṇita-yukti-bhāṣā, and (ii) C.T. Rajagopal and his collaborators, in a series of pioneering studies, drew attention to the significance of the results and techniques outlined in Yuktibhāṣā (and the work of the Kerala School of Mathematics in general), which seem to have been forgotten after the initial notice by Charles Whish in early nineteenth century. These and the subsequent studies have led to a somewhat different perception of the Indian contribution to the development of calculus as may be gleaned from the following quotation from a recent work on the history of mathematics:²

We have here a prime example of two traditions whose aims were completely different. The Euclidean ideology of proof which was so influential in the Islamic world had no apparent influence in India (as al-Biruni had complained long before), even if there is a possibility that the Greek tables of ‘trigonometric functions’ had been transmitted and refined. To suppose that some version of ‘calculus’ underlay the derivation of the series must be a matter of conjecture.

The single exception to this generalization is a long work, much admired in Kerala, which was known as Yuktibhāṣā, by Jyeṣṭhadeva; this contains something more like proofs—but again, given the different paradigm, we should be cautious about assuming that they are meant to serve the same functions. Both the authorship and date of this work are hard to establish exactly (the date usually claimed is the sixteenth century), but it does give explanations of how the formulae are arrived at which could be taken as a version of the calculus.

It has taken a long time for historians of mathematics to move from curiosity to uncertain admiration to well-informed scholarly recognition of the brilliance of the mathematicians/astronomers who lived and worked in Kerala (on the southwest coast of India) from the last quarter of the 14th century CE until the end of the 16th. As the undertaking of bringing to critical attention the totality of their work gathers pace, one thing has already become clear: it can no longer be doubted that the high point of their mathematical achievements is the invention of calculus and its systematic and sophisticated development for application to trigonometric functions. Three personalities have emerged as key figures in this story. The first of course is the still shadowy Mādhavan (Mādhavan Emprāntiri, or Mādhava in Sanskritised form; throughout this article, I employ the Malayalam way of writing names from Kerala, with a terminal n), the founder of the school, who is credited by his followers with having created much of their strikingly original mathematics and whose creative genius pervades everything that they subsequently did. Then we have the polymath Nīlakaṇṭhan (Nīlakaṇṭha Somayāji), the pivotal link between Mādhavan and the later generations and the author of a large body of surviving work, notably Tantrasaṃgraha (TS from now on). And the third is Jyeṣṭhadevan who wrote what can accurately be called the first textbook of calculus, Yuktibhāṣā (YB from now on). And the third is Jyeṣṭhadevan who wrote what can accurately be called the first textbook of calculus, Yuktibhāṣā (YB from now on). YB is a comprehensive account, in its last two chapters, of the fundamental principles of integral and differential calculus (in that natural order in the Kerala approach to calculus), as well as the relationship between them, and their use in the study of trigonometric functions.

Āryabhaṭīya (c.499) is the first extant text on mathematical astronomy in India to discuss the explicit algorithms for the computation of planetary positions [1]. Eccentric/epicyclic models are somewhat implicit in these algorithms in Āryabhaṭīya, and in the texts which followed. In contrast, the planetary theory in Ptolemy’s Almagest [2] and the later Islamic works which were influenced by Ptolemy, as also the Copernicus’s de Revolutionibus [3], are based on explicit geometrical models. In essence, the motion of any planet is viewed as a combination of uniform circular motions in these models. No such rigid principles are insisted upon in Āryabhaṭīya or even in the later Indian texts. Though most texts do not discuss the geometrical models at all, there are exceptions. Nīlakaṇṭha Somayājī who introduced a major revision of the traditional Indian planetary theory in his Tantrasaṅgraha [4, 5, 6], outlined a geometrical model of planetary motion in his other works, Golasāra, Siddhāntadarpaṇa and Āryabhaṭīya-bhāṣya. In this model, planets move in eccentric orbits inclined to the ecliptic around the mean Sun, which in turn goes aound the earth [7, 8]. We have a detailed account of this model in Gaṇita-Yukti-bhāṣā, popularly known as Yukti-bhāṣā (c.1530), authored by Jyeṣṭhadeva [9]. Here onwards, [9] will be indicated by just Gaṇita-Yukti-bhāṣā in all the specific references to the passages in it.