Panini And Euclid Reflections On Indian Geometry

Author(s): Johannes Bronkhorst
Publisher: Johannes Bronkhorst

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JOHANNES BRONKHORST
PĀŅINE AND EUCLID
It is not metaphorical-Punktchen is not trying to insinuate anything about the grandmother-like (or wolf-like) characteristics of her dog. But neither is it literal, and Punktchen knows this. Make-believe is a tertium between literality and metaphor: it is literality, but an as-if kind of literality. My theory is that the Greek diagram is an instantiation of its object in the sense in which Pinktchen's dog is the wolf - that the diagram is a make-believe object; it is functionally identical to it, what is perhaps most important, it is never questioned... The text) does not even hint whar, ultimately, its objects are; it simply works with an ersatz, as if it were the real thing... Undoubtedly, many mathematicians would simply assume that geometry is about spatial, physical objects, the sort of thing a diagram is. Others could have assumed the existence of mathematicals. The centrality of the diagram, however, and the roundabout way in which it was referred to meant that the Greek mathematician would not have to speak up for his ontology."
Whatever the Greeks may have thought (or not thought about the objects of geometry, geometry came to be looked upon as dealing with abstractions so much so that the expression "Euclid myth" has been coined to designate the false conviction that these objects have anything to do with the outside world: "What is the Euclid myth? It is the belief that the books of Euclid contain truths about the universe which are clear and indubitable. Starting from self-evident truths, and proceeding by rigorous proof. Euclid arrives at knowledge which is certain objective and eternal Even now, it seems that most educated people believe in the Euclid myth. Up to the middle or late nineteenth century, the myth was unchallenged. Everyone believed it. It has been the major support for metaphysical philosophy, that is, for philosophy which sought to establish some a priori certainty about the nature of the universe." (Davis and Hersh, 1981: 325, cited in Srinivas, 1990: 81 n. 14). For a recent discussion of the objects of (modern) mathematics, see Shapiro, 1997. "Cp. Aryabhatiya Bhasya p. 48 II. 15-16andha yatacaturasratvat tribhujasya.
The procedure illustrated in fig. 2 is close to the one which the Chinese mathematician Liu Hui justifies with the reason "Use the excess to fill up the void" (Chemla, 1999: 96 ff.). Chemia (1997) distinguishes between formal proofs (in italics) and proofs that are provided in order to understand the statement proved, to know why it is true and not only that it is true" (p. 229). * Aryabhatiya Bhasya p. 72 11. 10-17: athasannaparidhih kasmad eyale, na punah sphutaparidhir evocate' evare manyante: s updya evandstiyena siksmaparidhir dnlwate/ nanu cayam ast: wikkhambhavaggadasagunakarani vartassa parirao hodi (viskambhavargadafagunakarani vrasya parindho bhavati) ir atrapi kevala evdgamah naivopapanti paviskambharya dasa karanyah paridhir iti otha manyate pratyakenaiva pramilyamdno rüpaviskambhaksetrasya paridhir dasa karanya itinaitar. aparibhasitapramanarvde karaninam ekarrivistarayamdytacaturasrakerrakarena datakaranikenaiva tad viskambhaparidhir vestyamanah sa tatpramano bhavatiti cer tad api sddhyam eval. For the use of karani, see the reference in note 24 above. Note that Bhaskara's edited text has a number of grammatically incorrect occurrences of karanika- and karanitva: cp. Pån 7.4.14 na kapi.
The saying is in Prakrit and has obviously been borrowed from a Jaina text or context
I David Pingree (private communication) suggests that Bhaskara's problem was that no papatti could verify that = 10 because of the difficulty of relating the Square root of a surd to any previously verified theorem".
Hayashi's (1994: 123) following remark is therefore to be read with much caution: "The recognition of the importance of proofs dates back at least to the time of Bhaskara 1.... who, in his commentary on the Aryabhatiya, rejected the Jaina value of t. VIO, saying that it was only a tradition (agama) and that there was no derivation of it."
61 Aryabhatiya Bhasya p. 59 1. 3: ... Ini trairdsikopaparripradarsanartham ksetranycsah.
Aryabhatiya Bhasya p. 74 1. 9 - p. 75 1. 4: prsthanawanam api cu datakaran paridhiprakriyaparikalpanayd sada na (bhavall/ yalah) prsthanayane sutrami āryārdham: jyapadafarardhayutih svaguna (dasasanguna karar yas tahl Jatroddešakah: dvipancasadviskambhe dvir avagáhya "ogahanam vikkhambham" ir anena jyā labdhā vimsanih/20V (anaya jyaya) prsthanayanam: jpddah 5, Sarardham (I). yutih 6, svaguna 36, dadasariguna 360, etā karanyah prstham' sakalajydvargar carwari Satäni, prstham karaninām sastiçaratrayam ini, katham etar samghatate? wiyasi jatah prsthena bhavitavyum tad etad vicarymdnam aryantasāksmavadindm yitah prstham alpiyojmanam apatitam/ ato 'syai avicaritamanoharāyai namo's dafakaranya/. The edition has alpīyamanam. 45 The rule - which reads in full: ogāhānam wikkhambham egahena samgunam kurydd caügunlassa tu malam jivā savvakhattanam/l - is similar to Padalipta's Jyotiskarandaka 191, which has: ogahanam vikkhambha mo tu ogahasamgundum kujja/ caruhi gunitassa malam sa javd va'etha nätavall. 66 The Tattvarthadhigama Bhasya of Umasvati on sätra 3.11 (1 p. 258 II. 17-18) contains a different rule for the arc: "The arc (a) is the square root of six times the square of the sagitta (s) plus the square of the chord (c) (a = V(68' + c))" (isuvargasya sadgunasya jyavargayutarya mulam dhanuh kastham); later authors (Mahavira, Aryabhata II) accepted again different rules. See Datta, 1929: 694. 699. * For the approximation proposed by Heron of Alexandria, see Heath, 1921: II: 331. ** He could hardly have such a formula in view of his conviction that there is a chord equal to the arc (which it subtends": see above. " Michaels (1978: 56) points out that many terms related to the layering of the altar (the background of Vedic geometry) are known to Panini. In a note he mentions, or refers to, istakā, istakacit, aghicit, asadha, asvini, vayasa, etc. 70 Michaels's (1978: 70 ff.) attempts to show that the Sulba Sūtras contain theo retical statements about ideal objects ("theoretische Sätze über ideale Gegenstände") have to be treated with much caution, in the light of what we now know about classical Indian geometry and the geometry in other cultures (see below). About the relationship between Sulba Sutras and classical Indian geometry, see Kaye, 1919: 3 ("Les oeuvres de la seconde période (= Aryabhata etc.) ne font aucune allusion à un seul de ces sujets des Sulvasūtras") and the qualification added by Michaels (1978: 106: "Allerdings sind insbesonders angesichts dessen, dass einige Termini der [Sulba Satras] in der jüngeren indischen Mathematik, wenn auch mit teilweise neuer Bedeutung fortleben, hier gewisse Einschränkungen zu erheben"), " This does not withhold him from stating (Staal, 1999: 113): "The only Indian counterpart to Euclid is the derivational system of Panini's Sanskrit grammar." 72 To the publications mentioned by Staal one might add Michaels, 1978: 96 ff., and the literature referred to on p. 97 n. l. 7 Staal does not appear to address the question why in India - which purportedly had had both Påninian grammar and Euclid-like geometry - only the former came to play an important role in science and philosophy. Note in this connection that Aryabhatiya Bhasya p. 131. 24-p.16 1. 24 contains a long discussion purporting to show the greater importance of the study of Jyotisa than that of grammar 74 Reflections about the origin(s) of mathematics may have to take into consideration the extent to which mathematical activities were and are present outside the "higher" cultures, in societies without writing. See in this connection Marcia Ascher's book Erinomathematics (1991); further Ascher, 1994 7 Cp. also Friberg, 1990: 580: "There are reasons to believe that Babylonian mathem