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A NOTE ON ZERO AND THE NUMERICAL PLACE-VALUE
SYSTEM IN ANCIENT INDIA
Johannes Bronkhorst, Lausanne
The second volume of the Kalātattvakośa, which has recently (1992) come out, refers to zero and the numerical place-value system in its section called sünya: mathematical aspect" written by S.R. Sarma. Sarma is of the opinion that there is enough indirect evidence to say that the decimal place-value system with symbols for 1 to 9 and zero developed in India much before the beginning of the Christian era" (p. 403). This evidence is, according to Sarma, constituted by a passage in the Chandah-Sutra of Pingala, and by another one in the Jaina canonical text Anuyogadvāra.
Chandah-Sutra 8.28-31, Sarma points out, uses the symbols 2 and 0 in a meaningful way. He concludes: ,,The fact ... that Pingala uses them goes to show the existence of a well-recognised symbol for zero at his time, which is variously placed between 400 to 200 B.C." The problem here lies with the date he ascribes to the Chandah-Sutra. Various authors have pointed out that this date is not all that certain. This is what Renou (194749: 307) had to say about it: ,,Les Sūtra de Pingala sur la métrique, les Chandahsūtra, ... sont récents puisqu'ils traitent de mètres purement classiques." Note further that the passage referred to by Sarma does not even occur in the portion of the text dealing with Vedic metres, which some claim to be older. It will be clear that Sarma's chronological conclusions are not justified by the use of zero in the Chandah-Sūtra.
Regarding the Anuyogadvara, Jarl Charpentier (who refers back to Albrecht Weber) pointed out long ago (1914: 29) that it contains a list (section 49) which refers to the Vaiśeşika, Samkhya and Lokāyata systems of philosophy. Of these the Vaiśesika system is, according to Charpentier, the most modern, and indeed there is no evidence that this system existed before the beginning of our era. The Samkhya philosophy is referred to by the name Şaştitantra, which appears to have been the name of a work composed by Varşaganya, who may have lived in the beginning of the 4th century of our era (Frauwallner, 1958:131 (270)). Charpentier considers it ,,possible that Şastitantra is here only a name for the Sankhya system of philosophy, which is one of the very oldest amongst the Hindu
CF VAN NOOTEN, 1993: 33: ..... nor is it possible to prove that Pingala's work existed before the third century A.D."
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JOHANNES BRONKHORST
philosophical schools, being mentioned already by Kautilya" (p. 28), but this position is not tenable, for various reasons. It is not at all certain that Samkhya as a system is all that old; no evidence to that extent seems to exist. Kautilya's Arthaśästra is known to be a composite work, which reached completion long after the period of the Mauryas.2 And even if the Samkhya system were to be old, it is never referred to in early texts as Şaşţitantra. This is not without reason, for the term Şaşţitantra refers to the sixty principles which, if Frauwallner (1956: 320) is to be believed, did not enter the system until a late date.
A late date for the Anuyogadvara - or at least for this passage of it - is confirmed by the reference to a Pätañjali(ya) in it. Charpentier (p. 29) observes: "What Pätañjali(ya) means is doubtful; it may refer to the Mahabhäşya-vyäkaraṇa, however, is specially mentioned or rather to the Yoga-sütra's, which are certainly a late work." What Charpentier did not yet know, is that the terms Patanjali and Patañjala are used in the earlier literature with reference to Yoga Sutra and Bhasya together, never with reference to the Yoga Sutra alone (Bronkhorst, 1985: 203 ff.). Patanjali(ya) looks therefore even more recent than Charpentier realized.
It appears, then, that the evidence for a mathematical zero in India before the beginning of our era is very weak indeed. I should mention here in passing the recent publication by M.D. Pandit, Zero in Panini (1990).. Pandit is of the opinion that there is a resemblance between the mathematical zero and certain techniques used by Panini. Panini lived most probably in the fourth century before our era (Hinüber, 1990: 34). Does this imply that the mathematical zero existed before Panini, and therefore well before our era? Pandit is careful to avoid this conclusion (p. 116): "It is also possible that the ancient Indian mathematicians might also have borrowed certain techniques from Panini." We reach therefore the same conclusion as above: there is no solid evidence to believe that zero and the numerical place-value system existed in India before the beginning of our era.3
For the more recent period Sarma refers to a number of texts which testify to the use of zero and a decimal place-value system: Aryabhatiya, Pulisasiddhanta, Pañcasiddhantikā, Bṛhatkşetrasamäsa, Siddhasena Gani's commentary on the Tattvärtha Sutra, the Yoga Bhasya and Sankara's commentary on the Brahma Sutra. Sarma does not mention a passage in
2 See Asiatische Studien / Études Asiatiques 45 (2), 1991, p. 214 f.
3 VAN NOOTEN (1993:42-44) draws attention to the inscriptions of Naneghat (ca. 1st century B.C.E.), but admits that they do not contain a pure place-value system.
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A NOTE ON ZERO
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Vasubandhu's Abhidharmakośa Bhăşya (5.26; p. 296,1.21-23), referred to by Ruegg (1978:172-73). This passage is part of an opinion ascribed to a certain Bhadanta Vasumitra, and illustrates his position with the help of a marker or counter (vartikä) which in the unit position has the value of a unit, in the hundred's position that of a hundred, and in the thousand's position that of a thousand. Ruegg considers it probable that Vasumitra is "to be identified with one of the leading figures at the time of Kaniska's Great Council" and concludes that he could be the contemporary of Nāgärjuna (if not older)" (p. 175).
This conclusion is confirmed by the presence of the same - or a closely similar - passage in several earlier texts. It occurs in both the Chinese translations of the Mahāvibhāsā (T. 1545, vol. 27, p. 396b 1.3-4; T. 1546, vol. 28, p. 295c 1.18-20) and in the Vibhasa (T. 1547, vol. 28, p. 466b 1.18-19).4 The Mahavibhāşă may have been composed during the reign of Kaniska, and the Vibhāşă may be slightly older (Nakamura, 1980:107). It is therefore safe to conclude that the decimal place-value system was known in India during the early centuries of our era.5
REFERENCES AND ABREVIATIONS
Anuyogadvåra. In: Nandisuttam and Anuogaddaraim. Ed. by Muni Punyavijaya, Pt. Dalsukh Mälvania, Pt. Amritlål Monanlal Bhojak. Bombay: Shri Mahavira Jaina Vidyalaya. 1968. (Jaina-Agama-Series No. 1.) BRONKHORST, Johannes (1985): „Patanjali and the Yoga Sútra.“ Studien zur Indologie und Iranistik 10 (1984): 191-212. CHARPENTIER, Jarl (1914): The Uttarādhyayanasitra, being the first Mülasttra of the Svetämbara Jains, edited, with an introduction, text, critical notes and a commentary. First Indian edition. New Delhi: Ajay Book Service. 1980. FILLIOZAT, Jean (1953): „Chapitre IX. Les sciences." In: L'Inde Classique. Manuel des études indiennes. By Louis RENOU and Jean FILLIOZAT. Tome II. Paris: École Française d'Extrême-Orient. 1985. Pp. 138-194.
4
5
It also occurs in Sanghabhadra's Nyáyānusára (T. 1562, vol. 29, p. 631a 1.26), the Yoga Bhăşya (3.13), and several later works (see RUEGG, 1978:173-174). JOSEPH (1991:241) remarks: "If ... the original version of the Bakhshali Manuscript dates from the third century AD, it would be the earliest evidence of a well-established number system with a place-value scale and zero which is also recognizably an ancestor of our present-day number system." However, L'Inde Classique characterizes this manuscript as "relativement tardif", although "d'abord considéré à tort comme très ancien" (FILLIOZAT, 1953:175).
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________________ 1042 . . . - JOHANNES BRONKHORST FRAUWALLNER, Erich (1956): Geschichte der indischen Philosophie. 1. Band. Salzburg: Otto Muller. FRAUWALLNER, Erich (1958): ,,Die Erkenntislehre des klassischen Samkhya-Systems." Wiener Zeitschrift fur die Kunde Sud- und Ostasiens 2, 84-139. Reprint: Kleine Schriften (Wiesbaden, 1982) pp. 223-278. HINUBER, Oskar von (1990): Der Beginn der Schrift und fruhe Schriftlichkeit in Indien. Stuttgart: Franz Steiner Verlag Wiesbaden. (Akademie der Wissenschaften und der Literatur, Mainz. Abhandlungen der Geistes- und Sozialwissenschaftlichen Klasse, Jahrgang 1989, Nr. 11.) JOSEPH, George Gheverghese (1991): The Crest of the Peacock. Non-European roots of mathematics. London - New York: I.B. Tauris. NAKAMURA, Hajime (1980): Indian Buddhism. A survey with biographical notes. Osaka: KUFS Publication. (Intercultural Research Institute Monograph, 9.) PANDIT, M.D. (1990): Zero in Panini. Poona: Centre of Advanced Study in Sanskrit, University of Poona. (Publication of the Centre of Advanced Study in Sanskrit, Class B, No. 12.) PINGALA: Chandah-Sutra. Edited, with the commentary of Halayudha, by Pt. Visvanatha Sastri. Calcutta: Ganesa Press, 1874. (Bibliotheca Indica, New Series nos. 230, 258, 307.) RENOU, L. (1947-49): ,,Chapitre V. Le vedisme." In: L'Inde Classique. Manuel des etudes indiennes. By Louis RENOU and Jean FILLIOZAT. Tome I. Paris: Ecole Francaise d'Extreme-Orient. 1985. Pp. 270-380. RUEGG, David Seyfort (1978): ,,Mathematical and linguistic models in Indian thought: the case of zero and sunyata." Wiener Zeitschrift fur die Kunde Sudasiens 22, 171-181. SARMA, S.R. (1992): ,,Sunya: mathematical aspect." In: Kalatattvakosa, a lexicon of fundamental concepts of the Indian arts. General editor: Kapila Vatsyayana. Vol. II: Concepts of space and time. Edited by Bettina Baumer. Indira Gandhi National Centre for the Arts, New Delhi, and Motilal Banarsidass, Delhi. Pp. 400-411. T. = Taisho edition of the Buddhist Tripitaka in Chinese. VAN NOOTEN, B. (1993): ,,Binary numbers in Indian antiquity." Journal of Indian Philosophy 21, 31-50. Vasubandhu: Abhidharmakosa Bhasya. Edited by P. Pradhan. Revised second edition by Aruna Haldar. Patna: K.P. Jayaswal Research Institute. 1975. (Tibetan Sanskrit Works Series Vol. 8.)