Panini And Euclid Reflections On Indian Geometry

Author(s): Johannes Bronkhorst
Publisher: Johannes Bronkhorst

Page 14

JOHANNES BRONKHORST
PĀŅINI AND EUCLID
and mathematical literature. This may be brought out by compuring the grammar of Sanskrit with the geometry of Euclid - a particularly apposite comparison since. whereas mathematics grew out of philosophy in ancient Greece, it was, as we shall sec. partly an outcome of linguistic developments in India. The geometry of Euclid's Elements starts with a few definitions, axioms and postulates and then proceeds to build up an imposing structure of closely interlinked theorems, cach of which is in itself logically coherent and complete. In a similar fashion, Panini began his study of Sanskrit by taking about 1700 basic building blocks - some general concepts, vowels and consonants, nouns, pronouns and verbs, and so on- and proceeded to group them into various classes. With these roots and some appropriate suffixes and prefixes, he constructed compound words by a process not dissimilar to the way in which one specifies a function in modern mathematics. Consequently, the linguistic facility of the language came to be reflected in the character of mathematical literature and reasoning in India."
Maha-bh 1 p. 6 II. 3-7: katham tarkime sabdeh pratipattavydh kimeit suaminyavisesavar laksanam pravartyam yendipena yatena mahato mahatah sabdaughan pratipadyerukim puntas far utsargāpavadaw karcid utsargah karfayah kascid apavidal karhamjarvakah punar utsargah kartayyah kathamjatiyako pavidal samarrenotsarga kartavah. "Cp. Prakash Sarasvati, 1986: 157: "for the first time we find Aryabhata ... in his Aryabhatiya describing (mathematics) as a special section (Ganitaplida). Brahmagupta ... also followed Aryabhata in this respect and gave the science of calculation Ceanita) a special place in his treatise on astronomy. The Siddhanta treatises earlier than those of Aryabhata and Brahmagupta do not contain a chapter exclusively devoted to canita (the Sürya-Siddhanta and the Siddhantas of Vasistha, Pitamaha and Romaka are thus without ganita chapters)," Also Hayashi, 1995: 148: "Bhaskara (A.D. 629) tells us that Maskuri, Purana, Mudgala and other acaryas (teachers) composed mathematical treatises, but none of them is extant now. We can only have a glimpse of Indian mathematics of those early times through the extant astronomical works of Aryabhata, Bhaskara, and Brahmagupta, only a few chapters of which are devoted to mathematics proper. It should of course not be forgotten that much geometry is to be found in texts dealing with mathematical astronomy.
Aryabhatiya Bhasya p. 44 L 18: gamitam dviprakaram: rafiganitom ksetraganitom Cp. Hayashi, 1995: 62.
Aryabhatiya 2.17ab yas caiva bhajavargah kotivargas ca karnevargah sah. # Filliozat, 1988: 255-256. Another way of looking at the same characteristic is expressed in Pingree, 1978: 533, which speaks about the corrupt tradition of the earliest surviving Sanskrit texts: "The cause of this corruption is usually that the texts had become unintelligible; and this unintelligibility is not unrelated to the style developed by Indian astronomers. The texts proper were composed in verse in order to facilitate memorization, with various conventions for rendering numbers into metrical syllables. The exigencies of the meter often necessitated the omission of important parts of mathematical formulas, or contributed to the imprecision of the technical terminology by forcing the poet to substitute one term for another...."
Filliozat, 1995: 46 mentions the "high degree of generality" - which includes a general formulation of the Pythagorean theorem. "the first theorem enunciated in the history of Sanskrit mathematics" (p. 48) - as "a quality which a Sulbastra shares only with Panini model * Sharma, 1966, vol. III p. 829 Brāhmasphutasiddhanta 12.24) karnakse kotiktim risodra milam bhaje bhujasya krti prokya padam kotih kotibihurtytipadam karnahl. Tr. Colebrooke, 1817: 298. As a matter of fact, the Pythagorean theorem occurs several times, in several guises, in this portion of the text; e.g. 12.22cd: svavadhavargonad bhajavargdn mülam avalambak (tr. Colebrooke: "The square
root, extracted from the difference of the square of the side and square of its corresponding segment of the base, is the perpendicular"): 12.23cd: karakrtir bhimukhaytidalavargond padam lambah (tr. Colebrooke: "In any tetragon but a trapezium.) subtracting from the square of the diagonal the square of half the sum of the base and summit, the square root of the remainder is the perpendicular): 12.42ab: wdvysakriviesde milavyasantarardham isur alph (tr. Colebrooke: "Half the difference of the diameter and the root extracted from the difference of the squares of the diameter and the chord is the smaller arrow")
Bronkhorst, forthcoming 2: $ 2. "An exception may have to be made for the examples acompanying the sotras of the Bakhshall Manuscript
This is the name here adopted for the combined text consisting of the Daagliksetra vyakhya and the Aryabhata tantra-bhasya, expressions apparently used by Bhaskara himself to designate the two parts of his commentary (on chapter 1 and on the remaining chapters respectively see Shukla, 1976: xlix. 20 The Ganitaplida of Bhaskara's Aryabhatiya Bhasya has recently been studied and translated by Agathe Keller (2000). Reading her thesis has inspired me to write this article and helped me in the interpretation of many passages. a1 It is known that a number of classical commentaries imitated the style of the Mahabhisya, sometimes calling themselves "Vårttika" (Bronkhorst, 1990). Bhaskara's Bhasya does not adopt this so-called "Varttika-style" 22 Aryabhatiya Bhisya p. 96 L 15: vas cabhudargah war ca knivargah raw wargam ekatra karnavargo bhavari. 23 In order to avoid misunderstanding it must here be emphasized that I am aware of the fact that proof (Euclidean or other) does not appear to be essential to geometry in many cultures, see below. The present discussion on the absence of proof in Bhaskara and elsewhere finds its justification in the comparative approach adopted in this article, inspired by the remarks of Ingalls and Staal cited in its first pages. 24 Aryabhatiya Bhasya p. 44 L 17: kamabhujawah sematem karutisma atah karan, tr. Hayashi, 1995: 62. The dual is strange, and one wonders whether the original reading may not have been: bhujayoh karnasamavam karoti yamd fatah karanl. For a discussion of the term karanl in various mathematical texts, see Hayashi, 1995: 60-64 * Ganitapada 17cd: vitte sarasamvargo'rdhajyavargah sa thaly dhanusoh,
Aryabhatiya Bhasya p. 103 II. 12-13: pratyayakaranam ca sarvesy eva ksetresu "yas caiva bhudargah ko vargas ca karnawargah sah" iry anenaiveti. 27 Earlier authorities have already drawn attention to the absence of proofs in early classical Indian mathematics. So eg. Kline, 1972: 190, as cited in Srinivas, 1990: 76 n. 2: "There is much good procedure and technical facility, but no evidence that (the Hindus) considered proof at all." See however below. - Parallel to the hypothesis here considered to the extent that Panini may have exerted a negative influence on the development of certain sciences in India, is Lloyd's (1990: 87 ff.) observation that the influence of Euclid's Elements on the development of Greek science was not only positive. Lloyd draws attention in particular to medicine and physiology, certain areas of mathematics itsell, and to the extent to which problems in physics and elsewhere had a tendency to be idealized exactness may be obtained only at the cost of applicability)
This enumeration is not exhaustive even with regard to pusages identified in the text. Passages from the Mahabhasya are identified p. 3 II. 7-9, p. 8 II 1-2, p. 23 11. 25-26; but they are not mentioned in the appendix
It is tempting to see furthermore, in the line luke cu me i ganitapokaruh "wam wrddhimako 'pacaydimakova na bhavuti (p44 II 6-7, introducing Gunitapada 1) a reflection of Vkp 1.131ab: na so si pravayo lake yah kahdanwgamad nie.
62 The carattu jolie and Staaletines are
kative in pothesis fera
(1990: 32