Panini And Euclid Reflections On Indian Geometry

Author(s): Johannes Bronkhorst
Publisher: Johannes Bronkhorst

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JOHANNES BRONKHORST
PANINI AND EUCLID
Bhaskara uses this diagram to show that the square of a diagonal of a rectangular surface does indeed correspond to a geometric square: AC (AB + BC) corresponds to the surface ACDE. However, this same diagram could easily be used, and has been used by more recent authors, to prove the Pythagorean theorem (though not, of course, in a "Euclidean" manner). In this case, as Keller points out in her thesis, the area of the interior small square (whose sides are equal to b-a) increased by the area of the four triangles (whose sides are a and b). gives the area of the big square whose sides are the hypotenuse of the four triangles in other words: a = (-a) + 4 x ab/2 = 22 +b?. This proof is given by the author Ganesa Daivajna in around 1545 (Srinivas, 1990: 39) and, it is claimed, by Bhaskara Il around 1150. Bhaskara I had this proof of the Pythagorean theorem so to say under his nose, but he was apparently not interested.
We are therefore led to the following conclusion. There is no proof for the claimed methodical guidance of Panini's grammar with respect to classical Indian geometry, except perhaps where completely external features of presentation are concerned. Contentwise, classical Indian geometry contrasts as sharply with Euclidean geometry as do other pre-modem geometries - in China, Babylonia and Egypt - which had no knowledge of Panini's grammar. Like these other pre-modem geometries, classical Indian geometry did not use proofs. The most noteworthy distinguishing feature of classical Indian geometry is that, unlike the other pre-modem geometries, it developed in surroundings where the notion of proof was well-established. It is not immediately clear whether this lack of susceptibility to the notion of proof on the part of classical Indian geometry is in need of an explanation. If it is, Päņini's grammar might conceivably be enumerated among the factors that played a role. In that case one may have to consider the possibility that the influence of Panini's grammar, far from encouraging the development of an abstract geometry, had the opposite effect.
NOTES
Where does all this leave us with regard to the influence of Panini's grammar on geometry in India? Those who, like Staal and others, are inclined to look upon Panini as having provided methodical guidance to the Indian sciences, will find features in Aryabhata and Bhaskara that remind them of Panini's grammar and which they might like to attribute to the latter's influence on the former. If so, they may have to consider whether and to what extent this influence is responsible for the absence of proofs in early classical geometry. A comparative study of various ancient cultures shows that proofs in geometry do not normally appear, unless social, political or other circumstances have led to the kind of awareness in which an explicit notion of proof has found its place. Such a notion of proof did exist in classical India at the time of Aryabhata and Bhaskara, but among philosopher-logicians, not among mathematicians. This peculiar situation may in part have to be explained by the fact that mathematicians were less exposed to debate and controversy than the philosophers. To this must be added that the Indian mathematicians may have been happy with their methods which - if not in the case of Aryabhata and Bhaskara, but certainly in that of Brahmagupta and others - led to remarkable results. This does not change the fact that the example of the grammar of Panini may conceivably have lent added respectability to a geometry without proof, even at a time when mathematicians came across this notion (whether under Greek mathematical or Indian philosophical influence). Mathematics in the style of Aryabhata and Bhaskara had to be good enough, for it resembled in some essential respects Panini's grammar, which certainly was good enough.
This article has been inspired by the recent thesis of Agathe Keller (2000). I have further profited from comments on an earlier version by Pascale Hugon, David Pingree and Kim Plofker. The responsibility for the opinions expressed and for mistakes remains mine.
Cp. further note 9, below. Filliozat (1995: 40) observes: "Obviously in the mind of the Indian learned men the study of language held the place which mathematics held in the mind of the ancient Greek philosophers." See also Ruegg, 1978.
Cp. Pingree, 1981: 2: "classical astronomy and mathematics had virtually ceased to be studied or taught by the end of the nineteenth century. A new group of Indian and foreign scholars has, however, begun to work in these areas since World War II".
The reference is to B.V. Subbarayappa and K.V. Sarma, Indian Astronomy: A Source Book, Bombay 1985.
Note further that "lalocording to Pingree's estimation ..there still exist today some 100,000 Sanskrit manuscripts in the single field of jyntialästra Gastronomy astrology, and mathematics)" (Yano, 1987: 50).
On Indian medical literature see, in particular, Meulenbeld, 1999 ff. * I am aware of two exceptions. One is Singh, 1990, which tries to show, not very convincingly, that algebraic inclination of ancient Indian mathematics was strongly related to foundational attitude developed by linguistic tradition" (p. 246). The other exception is Filliozat, 1995.
For some reflections on the limited influence exerted by Panini's grammar on philosophy in India, see Bronkhorst, forthcoming.
On the Greek influence on Indian astronomy, see Pingree, 1971, 1993. On the extent to which Greek geometry influenced Indian geometry, see also Shukla's remark cited in note 47, below.
Cp. Joseph, 1991: 217-219: "An indirect consequence of Panini's efforts to increase the linguistic facility of Sanskrit soon became apparent in the character of scientific