The Buddhivilasini: Commentary of Ganesa Daivajna on the Lilavati of Bhaskaracarya II

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THE Lilavati of Bhaskaracarya occupies a prominent place among the Sanskrit texts on arithmetic and geometry. With its crisp and precise rules and highly imaginative examples, the Lilavati became popular throughout India, as is evident from about 600 manuscript copies of this work, written in almost all the scripts in India, which are listed in David Pingree's The Census of Exact Sciences in India (vol. A4, Philadelphia 1981). Its popularity is also indicated by the numerous commentaries written on this work.

Outstanding among these commentaries is the Buddhivilasini of Ganesa Daivajna of Nandigrama in Konkan. In a Vasantatilaka verse at the end of this commentary, Ganesa states that he completed it on caitrasukla-pratipada, in the Jovian year Visvavasu, 1467 Saka, which translates to Saturday, 14 March 1545. Like Bhaskaracarya, Ganesa was also a brilliant scholar and poet. He authored several works on astronomy, calendar, astronomical instruments and other related topics. Notable among these works is the Grahalaghava which was likewise highly popular.

A conventional commentary explains the meanings of words and their grammatical formation, identifies the metres and the figures of speech, with appropriate citations from lexica and other texts. Since Bhaskara intended that his Lilavati should be understood even by children without much effort (balalilavagamya), Ganesa asserts that in preparing an ordinary commentary on such a lucid text does not involve any great skill; therefore, he presents more than 100 upapattis to the rules enunciated in the Lilavati and thus displays his intellectual. ingenuity (buddhicitra). Significantly, he named his commentary Buddhivilasint, "that which enlivens or enchants the reader's intellect". It should be added that the Buddhivilasint does not contain just upapattis; it also performs the normal functions of an ordinary commentary.

Upapatti, also called vasana or yukti, is not proof in the Euclidian sense, but rather a justification or rationale for the rules enunciated in the text. Bhaskara himself provides upapattis or vasanas in his auto-commentaries entitled vasanabhasyas on his Bijaganita and Siddhantasiromani.

These upapattis in the Buddhivilasini are the main focus of V. Ramakalyani's work The Buddhivilasini: Commentary of Ganesa Daivajna on the Lilavati of Bhaskaracarya II: A Critical Study of Proofs in Indian Mathemetics of Sixteenth Century, for which she was awarded the degree of Doctor of Philosophy by the University of Madras in 2018.

This work is divided into nine chapters. The first chapter gives an overview of mathematical texts in Sanskrit and introduces Bhaskara and Ganesa and their works. The next seven chapters discuss and analyse very systematically the upapattis presented in the Buddhivilasini on the successive chapters of the Lilavati.

In each case, Dr Ramakalyani cites the relevant rule or example from the Lilavati (either in translation or as paraphrase), reproduces the full text of the upapatti on it after correcting the misprints in the Anandashrama edition translates it meticulously into English, explains it in modern notation, oftentimes with suitable diagrams, and discusses it critically. Sometimes, she adds insightful remarks or notes on the significance of the upapatti concerned. All of this is done in an exemplary manner.

She notes that Ganesa employs different strategies in his upapattis. Some are verbal explanations of the procedure of computation, some others employ arithmetical means (vyaktaritya, vyaktakriyaya), others present algebraic proofs (avyaktaritya, avyakta-kriyaya) with citations from Bhaskara's Bljaganita, yet others provide geometrical proofs (ksetragatopapattya), sometimes with appropriate diagrams.

In the Lilavati, the chapter "Ksetravyavahara" dealing with plane geometric figures is the largest chapter. Accordingly, Ganesa's commentary on this chapter is also the largest and the more detailed. Ganesa commences his commentary with a classification of the figures. He observes that there are fourteen types of plane figures, comprising three types of triangles, ten types of quadrilaterals, the last being the circle and bow, which are treated as one type. Other figures like pentagons, etc. are treated as combination of two or more triangles.

The final chapter evaluates Ganesa's contribution to mathematics, his style as a commentator and his erudition. The author draws attention to the fact that Ganesa formulates some rules of his own both in verse and prose, viz. rules to determine sine and versine values, the area of a capaksetra (segment of a circle bounded by a chord and bow), the altitude of suci triangle and so on and that he adds also some examples of his own.

Ganesa displays his erudition by composing verses with triple or double meaning. The very first verse of the commentary, where he salutes his father and teacher Kesava, has three layers of meaning, all of which he explains meticulously: one pertaining to his father Kesava, another pertaining to Kesava Visnu and a third pertaining to the divine sun (prasasta kesa rasmayo yasyasau kesavah).

THE knowledge and development of technical science especially that of mathematics provides the parameter to assess the real phase of a civilization. India had made great strides in the field of mathematics as early as in the Vedic age by propounding place value system, large numbers, geometrical constructions and irrational numbers. However, it is only in the recent past that the eyes of the mathematicians of the world have turned towards the mathematical excellence of ancient India.

Ancient Indian works, be they part of the Vedas, Upanisads, the epics and the Puranas, are found in the form of simple aphorisms known as sutras (rules and verses) facilitating memorization thereby enabling easy propagation by word of mouth. This procedure has helped in presenting facts relating to science, astronomy, mathematics, encyclopaedic literature, etc. in a precise and concise form. Mathematics, being a discipline dedicated to clarity, precision and conciseness, has benefitted effectively by this method adhered to by Indian mathematicians of yore. As centuries passed by, the comprehension of these sutras became difficult; commentaries were written by the later mathematicians for easy understanding of these sutras. In the present times the critical study of these commentaries has become imminent to explain the original texts. The present book offers to critically analyse the Buddhivilasini of Ganesa Daivajna, a well-known and reputed commentary on the Lilavati of Bhaskara II. The first chapter being an introduction to the study provides the following details:

1. A brief account of the history of mathematics in ancient India,

2. Bhaskaracarya II and the Lilavati, and

3. Ganesa Daivajna, his works and the Buddhivilasini.

1.1 Indian Mathematics

The history of mathematics in India goes as far back as the Vedic times. The number system as enunciated by the Vedas is similar to the one that we are following throughout the world, today. The Vedic number system is a decimal number system, dividing the number ranks on the basis of 10 and its multiples. The value of the number is in terms of its place. In the texts of Vedas (2000-800 BCE), terms denoting powers of 10 up to 1012, are given by the Vajasaneyi-Madhyandina Sukla-Yajurveda Samhita (XVII.2). They are eka = 1, dasa = 10, sata = 102, sahasra = 103, ayuta = 10, niyuta = 10°, prayuta = 10º, arbuda = 10, nyarbuda = 10%, samudra = 10°, madhya = 1010, anta = 101" and parardha = 1012. An important result of the place value notation is zero, which is the greatest discovery in mathematics. It has revolutionized the mathematical knowledge in the world. History tells us that the Babylonian mathematics had a scale of sixty and the Mayan mathematics was based on twenty (Smith 1958: 35-52). Yet, in spite of the spread of these ancient scales, the Vedic number system based on ten surpassed them all as this facilitated easy operations on numbers. Mathematics in ancient India, even in Vedic times, has always served as the handmaid to other sciences like astronomy and astrology. The astronomical treatises like the works of Aryabhata I and II and Varahamihira employ mathematics to their purpose as a tool.

1.1.1 SULBASUTRAS

The Sulbasutras are part of the Kalpasutras, one of the six constituent parts of the Vedangas, which are considered as having special importance as far as mathematics is concerned, particularly geometry. These, as the earliest source of mathematics in India, specifically deal with rules for measurements and constructions of the various sacrificial altars. Consequently, they involve geometrical propositions and problems relating to rectilinear figures making their combinations and transformations, squaring the circle and circling the squares as well as arithmetical and algebraic solutions of problems arising out of such measurements and constructions. The word sulba means "a cord", "a rope" or "a string", and its root sulb signifies "measuring" or "act of measurement". Interestingly, Egyptians also considered geometry of surveying to be the science of the "rope stretchers" (harpedonap'tae) who appear to be the Egyptian counterparts of the Indian sulbavids (Sen and Bag 1983: 1-2).