Panini And Euclid Reflections On Indian Geometry

Author(s): Johannes Bronkhorst
Publisher: Johannes Bronkhorst

Page 15

JOHANNES BRONKHORST
PANINI AND EUCLID
Cp. however na so'sti puruso loke yo na kamayate Sriyam in Jagaliyotirmalla's Slokasárasamgraha, quated froun the Hitopadeša (Lindtner, 2000: 60, 65).
Many of the following quotations from the Mahabhasya could be identified thanks to the electronic version of that text prepared by George Cardona. No attempt has been made to identify all quotations from the Mahabhāsya.
Mahabh II p. 246 1 6: ciminyucudunds tu visesesv avansthante.
Maha-bh I p. 5 1. 28yah sarvatha ciram jīvari sa varsasalam jivati. * Maha-hh III p. 32 II. 4-5. 1 Muha-bh Il p. 437 1. 2. Maha-bh I p. 218 IL 15-16 (on Pan. 4.1.48 vt. 3) gives as example of târsthyar: manca hasanii, where Bhaskara gives maricah krocantl.
The Kasika on Pan. 6.4.11 has samasānto vidhir aniryak with variant reading sumisen tavidhir anityah
Compare Aryabhatiya Bhasya p. 7 H. 7-8 (etad ekalkasya granthalaksanalakayam maskaripunamadgalaprubhrtibhir dcärvair nibaddham krtam ...) with Maha-bh I p. 12 II. 15-17 (Laksyalaksane vyakaranam (vt. 14V laksyam ca laksanam caitar samuditam vydkaranam blavar kim punar laksyam laksanam ca sabido laksyah Surram laksanam/). W David Pingree (private communication) informs me that this employment of stra became common in commentaries on mathematical and astronomical texts. In other disciplines the word surra refers much less commonly to a metrical mula text, two texts that do so are the Yuktidipika (on the Samkhyakärika) and the Abhidharmakosabhasya W They can coincide with verses, in which case Bhaskaru, in the first chapter, uses the expression Ritikasura (or girikasarra, Ririsutra; e.g. Aryabhatiya Bhasya p. 1 II. 10-11, p. 7 II. 13 & 16. p. 11 II. 14 & 20, etc.). elsewhere dryasutra (eg p. 247 1. 20). # Aryabhatiya Bhasya p. 105 H. 12-17. p. 107 II. 10-11. 41 So Hayashi, 1994: 122: "Neither the Aryabhatiya nor the Brihmasphutasiddhanta contains proofs of their mathematical rules, but this does not necessarily mean that their authors did not prove them. It was probably a matter of the style of exposition." " D.E. Smith (1923: 158) claims to find faulty theorems in Brahmagupta's Brahmasphutasiddhanta, but most of his cases concern a rule (12.21ab) which is presented as approximate (sthala) by its author. His one remaining case - a formula for the area of quadrilaterals that is presented as being valid without restriction, but is in reality only valid for cyclic quadrilaterals (12.21cd) - may have to be interpreted differently: J. Pottage, at the end of a detailed study (1974), reaches the following conclusion (p. 354): "I have been unable to accept that Brahmagupta could have imagined that his rules would apply to all quadrilaterals whatsoever". 41 Aryabhatiya Ganitapada 6cd: ardhwabhujātaisam wardrdham sa ghanah sadurir HCP. Aryabhatiya Ganitapuda 7sumaparinahasydrilham wykambhardhahatam eva vrttaphaland tannijamalena haram ghanagolaphalam niravaresami "Half the even circumference multiplied by half the diameter is precisely the fruit (i.c., the area) of u circle. That (the area) multiplied by its own square root is the exact volume (lit. the without-a-remainder solid fruit) of a sphere." (tr. Hayashi, 1997: 198; similarly Clark, 1930: 27). Bhaskara also provides an approximate, practical" (vydvaharika), rule for calculating the volume of a sphere (p. 61 L 27): vydsdrdhagharam bhitra ravagunitum uygudasya ghanagaritam "Having divided into two the cube of half the diameter multiplied by nine, the calculation of the volume of an iron ball (has been carried out." * Filliozat and Mazars (1985) mention Conrad Müller (1940) and Kurt Elfering (1968, 1975, 1977).
See also Hayashi, 1997: 197-19. Smeur, 1970: 259-260 presents a hypothesis
sery DENONE
to explain how the incorrect rule for the volume of a sphere might have come into being. 47 Keller (2000: 188) suggests the following explanation: "Bhaskara semble considerer qu'il y a une continuité entre la figure plane et la figure solide. Cette continuite pourrait servir d'explication aux formules fausses de l'Aryabhatiya. Ainsi, la lecture du calcul sur le volume du cube repose sur la lecture du vers qui fournit l'aire du carré. Le volume du cube est le produit de l'aire du carré par la hauteur (V = A x H). De même le volume de la sphère est la racine carrée de l'aire multipliée par l'aire (V = AX VA). Le volume de la Sphere semble encore une fois etre le produit d'une aire et d'une «hauteur que représente numériquement la racine-carré de l'aire. L'aire d'un triangle équilateral est le produit de la moitié de la base et d'une hauteur (A = 1/2 B x H). Dans la continuité de cette aire, le volume du tétraèdre est donné avec la même pondération: la moitié de l'aire du triangle Squilatéral et de la hauteur (V = 1/2 A X H)." Similarly Plofker, 1996: 62: "This error... suggests that in this case reasoning by analogy led Aryabhata astray." Shukla (1972: 44) observes: "It is strange that the accurate formula for the volume of a sphere was not known in India. This seems to suggest that Greek Geometry was not known at all in India..." 4 Contrast the errors of Aryabhata and Bhaskara with the situation in classical Greece: "One of the most impressive features of Greek mathematics is its being practically mistake-free. An inspectable product in a society keen on criticism would tend to be well tested." (Netz, 1999: 216)
Aryabhatiya Bhasya p. 11 1 23 - p. 12 I. 1: anenācāryena mahadbhis fapobhir brahmaradhitah... / ato 'nena lokanugraha ya sphutagrahagatyarthavacakani dasa gitikasatrani ganitakalakriyagolarthavacakam aryasrasatan ca vinibaddham/. 50 The Greek situation tends to be idealized. A closer study of the evidence leads Netz (1999: 95) to the following assessment: "Greek mathematical works do not start with definitions. They start with second-order statements, in which the goals and the means of the work are settled. Often, this includes material we identify as "definitions'. In counting definitions, snatches of text must be taken out of context, and the decision concerning where they start is somewhat arbitrary."
Aryabhatiya Bhasya p. 77 IL. 9-15: vayam tu brūmah: asti kasrharulyajyeri yadi kästhalulyajya na syit tada samdyam avanda vyavasthanam evayogudasya na syat tenanumimahe kascir pradesah so sriti yenāsāv ayogudah samayam awanav avaristhate/ sa ca pradesah paridheh sannavalyumsal kaytharuluyatnyair apy ācāryair abhyavagata: tarparidheh Satabhagom sprati dharam golakasurirar iti. 12 The quoted line is problematic, not only because of the neuter ending of sarabhagam. but even more so on account of golakasarirar where one would expect something like golakasarirahvat. Shukla (1976: Ixiv) translates: "Due to the sphericity of its body, a sphere touches the Earth by one-hundredth part of its circumference": Hayashi (1997: 213) has: "A hundredth part of its circumference touches the ground because of its having a spherical body." » Kim Plofker reminds me that one cannot balance a circle on level ground because it would fall over sideways; no confusion may therefore be involved here. S4 Maha-bh I p. 71.28 - p. 81. 1: gharena karyam karisyan kumbhakarakulam gatvaha kuru ghatam karyam anena karisyamiri na fadvac chabdan pravoksvamänd vaiyakaranakulam gatvāla kuru sabdan prayoksya itil. 55 Maha-bh I p. 11 II. 1-2. * Concerning the nature of the objects of Greek geometry, read the following remarks by Reviel Netz (1999: 54-56): "On the one hand, the Greeks speak as if the object of the proposition is the diagram. ... On the other hand, Greeks act in a way which precludes this possibility ... Take Punktchen for instance. Her dog is lying in her bed, and she stands next to it, addressing it: 'Bur grandmother. why have you got such large teeth? What is the semiotic role of 'grandmother?