Panini And Euclid Reflections On Indian Geometry

Author(s): Johannes Bronkhorst
Publisher: Johannes Bronkhorst

Page 17

JOHANNES BRONKHORST
PANINI AND EUCLID
able for
were not avere (Vogel. 1997
fora
that role was not alcation to the thesis we seem a merely and plays a role
ties in a decisive way influenced Egyptian, Greek, Indian and Chinese mathematics. in form as well as in catent, during the last half of the -Ist mill., if not earlier."
Lloyd sums up his views on the development of proof in the following passages (pp. 95-96): "We can say that the development, in Greece, of the demand for cen y sprung in part from a dissatisfaction shared by a variety of individuals with the merely persuasive.... We have related other intellectual developments that took place in early Greck thought to the political and social background, for example the extensive experience that many Greeks had of evaluating arguments in the law courts and assemblies.... We should conclude that in the development of formal or rigorous proof too... the political and legal background plays a role at least at the beginning of what might otherwise seem a merely intellectual development. However, the qualification to the thesis that must be entered is that, in this instance, that role was not as a source of positive, but rather of negative, models." " Cp Friberg. 1990: 583: 't is clear that the Greek mathematicians completely transformed the intellectual goods they received [from Babylonian mathematics).
Rigorous proofs based on abstract definitions and axioms took over the role played in Babylonian mathematics by a conceptually simpler method, that of using a reversal of the steps in an algorithm with given numerical data in order to check the computed values."
G.G. Joseph - in a chapter called "Egyptian and Babylonian mathematics: an assessment" - proposes to adjust the notion of proof so as to include these traditions (1991: 127): "A modern proof is a procedure, based on axiomatic deductido, which follows a chain of reasoning from the initial assumptions to the final conclusion. But is this not taking a highly restrictive view of what is proof? Could we not expand our definition to include, as suggested by Imre Lakatos ... explanations, justifications and elaborations of a conjecture constantly subjected to counter-examples? Is it not possible for an argument or proof to be expressed in thetoric rather than symbolic terms, and still be quite rigorous If one is determined to find proofs in all cultures that had geometry or something resembling it, adjusting the notion of proof may he the way to succeed; it is however open to doubt whether such a procedure adds much to our understanding
It appears that a copy of a Chinese translation of the Elements was present in the imperial library at the end of the thirteenth century, but was ignored; see Needham, 1959. 105 ff. Huff (1993: 241; with a reference to Aydin Sayili, The Observatory in Islam, Ankara 1960, p. 189) adds: "Even more tantalizing are the reports that a Mongol ruler in China, Mangu (d. 1257 ...) is said to have mastered difficult passages of Euclid by himself
Engelfriet, 1998: 297-298
Martzloff, 1997: 275. Yabuuti (2000: 40) suggests that Chinese mathematicians deduced theorems like that of Pythagoras by analogy, intuitively.
Martzloff, 1997: 72
Note further Marteloff. 1997: 276:"certain texts by Liu Hui (end of third century C.E.) and other mathematicians contain reasoning which, while it is not Euclidcan, is no less well constructed and completely exact. Moreover, although they are not numerous, these arguments appear all the more salient because they are without peers in other non-Euclidean mathematical traditions. But they also enable us to understand that Chinese mathematics is in part based on a small number of heuristic operational methods of a geometrical type... In fact, the most striking thing is the concrete appearance of the Chinese approach or rather the fact that abstract results af ved via concrete means. Chinese proofs tend to be based on visual or manual illustration of certain relationships rather than on purely discursive logic."
Bronkhorst. 1999 "C. Collins, 1908: 551: "Organizationally, the mathematiciuns, astronomers, and medical distors were based in private familistic lineages and guilds, never part
of the sustained argument provided by philosophical networks. Public networks of argument did exist in India, its philosophical lineages reached high levels of abstract development Only mathematics and science were not carried along with it." The striking absence of a Buddhist contribution to and participation in the development of astronomy and mathematics in classical India may be partly responsible for the relative peace" enjoyed by these branches of learning. (Vogel, 1997, shows that the Buddhists - or at least some of them were not averse to following developments in astronomy to fix the dates of their Posadha ceremony. The need to fix these dates did not apparently have the same effect as the Christian need to fix the date of Easter: about this later need Duncan, 1998: 79 states: "The history of science in the Middle Ages would have been very different if the bishops at Nicaea had decided to name a fixed date for Easter in the solar calendar"; further details in Heilbron, 1999.) Note that votisa/jyotis is mentioned in some lists of kalds occurring in Buddhist texts see Franco, 2000: 550 ([61]) with note 56. WA whole chapter of Brahmagupta's Brahmasphutasiddhanta (no. 11: Tantrapariksa) is dedicated to the refutation of different opinions.
For a recent example (from Jainism) of the close link between calendrical (ie. astronomical) and sectarian concerns, see Cort, 1999
He calls his teacher saratantranatantra and sarvesu antresu samom svatantra his father advaitavidyācārya (CESS 5 (1994), p. 735). " The one quotation from the Vakyapadiya (p. 22) concerns the meaning and function of upaya. Be it noted that Bhaskara's commentary on Gitikäpada 1 enumerates the five buddhindriyas, the five karmendriyas, plus manas, buddhi and sharikara, all of them known from Samkhya (p. 4 II. 11-15). Bhaskara also shows some acquaintance with Mimams: p. 182 I. 8 sarvasākhapratyayam kam karma may have been cited from Sabara's Bhasya on Mimamsasutra 2.4.9: the discussion immediately preceding this has a parallel in Sabara on 1.3.2
Mathematics is most often presented in treatises of astronomy, and it seems likely that astronomers often cared their living as astrologers. Cp. Al-Biron (E.C. Sachau's translation as reproduced in Chattopadhyaya, 1992: 510): "If a man wants to gain the title of an astronomer, he must not only know scientific or mathematical astronomy, but also astrology." Cp. Pingree, 1981: 56 as cited in Yano, 1987: 54: "There was never in India a jati(caste, MY] of mathematicians, and rarely anything that could be called a school: most mathematicians were joriy (astronomers or astrologers)." Pingree (1993: 77) argues that Aryabhata, far from making observations himself, derived the longitudes of the planets "from astrological playing with numbers". "On the low esteem in which astrologers were held, see Kane, 1974: 526 ff.
Aryabhatiya Bhāsya p. 481. 16 ff. The edition of K.S. Shukla contains a somewhat different diagram; however, a manuscript page reproduced in Keller's thesis (1 p. 223) appears to support her construction " Cp. Srinivas, 1990: 35; Sarasvati Amma, 1999. 133 ff. Keller refers in this connection to an unpublished Ph.D. thesis of Simon Fraser University: A critical edition, English translation and commentary of the Upodghāta Şad vidhaprakarana and kutakadhikara of the Suryaprakasa of Süryadasa, by Pushpa Kumari Jain, 1995. " David Pingree (private communication) points out that certain theorems on cyclic quadrilaterals presented by Brahmagupla (628 C.E.) were not developed from a Euclidean approach until the 16th and 17th centuries in Europe.