Panini And Euclid Reflections On Indian Geometry

Author(s): Johannes Bronkhorst
Publisher: Johannes Bronkhorst

Page 12

JOHANNES BRONKHORST
PĀŅINI AND EUCLID
one's critics and defend one's own point of view. This is not the place to discuss the enormous impact which this tradition of rational inquiry has had on the shape and direction of Indian philosophy, but it is clear that the development of logic in the different schools of philosophy was a result of this ongoing confrontation. Logic specified the rules which even one's greatest enemy would have to accept.
It seems likely that those who studied and practiced mathematics in India were not to the same extent as their philosopher contemporaries obliged to defend their positions against opponents who disagreed with practically every single word they uttered (or wrote down). There were differences of opinion, to be sure, but they were apparently not looked upon as particularly threatening. Bhaskara could show that the Jaina value for, or rather their formula for calculating the length of are that used this value, could not be correct. But apparently such disagreements were not yet considered sufficiently serious to rethink the whole system. Indeed we have seen that Bhaskara does not mind citing an apparently Jaina rule to justify a calculation. Indian astronomers and mathematicians, it appears, were not engaged in any such ongoing debate with fierce opponents belonging to altogether different traditions as were the philosophers.7 Nor were they - it seems - particularly interested in what was going on in philosophy. Randall Collins claims (1998:551) that there are no recorded contacts between philosophical and mathematical networks, and that no individuals overlap both activities. This may not be entirely correct: David Pingree (private communication) mentions in this connection Nilakantha Somayajin's Jyotirmimämsä (written in 1504; cp. Pingree, 1981: 50. 128); and Jean Michel Delire - in a paper read at the Xith World Sanskrit Conference, Turin, April 2000 - has drawn attention to Venkateśvara Diksita, a late 16th century commentator who combined skills in Mimamsa and mathematics (Pingree. 1981: 6. 129).
Yet Collins's claim appears to hold true for Bhāskara (and perhaps other mathematicians of his time), judging by the list of authorities cited by him which is given at the end of the edition of his Aryabhatiya Bhasya (Shukla, 1976: 345-346). Bhaskara often cites grammatical and generally linguistic texts (as we have seen), astronomical texts, some religious and literary treatises, but not a single philosophical work. It is true that we have very little information about the lives and circumstances of the early Indian mathematicians, but there is no reason that I know of to doubt their relative intellectual isolation, combined perhaps with low social esteem. It goes without saying that further research in this complex of questions is called for.
Some scholars are of the opinion that more recent Indian mathematical authors did provide proofs for geometrical theorems. Takao Hayashi, for example, remarks (1995: 75): "the term upapatii stands for the proof or derivation of a mathematical formula. We find a number of instances of upapatti used in that sense in later commentaries such as Ganesa's Buddhivilasini (A.D. 1545) on the Līlāvati and Krsna's Navānkuru (ca. A.D. 1600) on the Bījaganita." M.D. Srinivas (1990) gives a list of commentaries that contain mathematical upapatris in an appendix (no. I; p. 57 ff.) to his article; all of them date from the 16th and 17th centuries. The highly interesting question to what extent these later mathematical authors had an explicit concept of proof (not necessarily Euclidean; cp. Lloyd's remarks, cited above) and, if so, when, how and why such a concept made its appearance in Indian mathematical works cannot be addressed in this article. One may wonder whether the type of arguing that had become common in philosophical debate slowly found its way into this area of knowledge. But whether or not such a shift of attitude took place in mathematics, there seems to be no doubt that Bhāskara I, the commentator whose work we are considering, was not (yet) affected by it.
The following example confirms this. The Aryabhatiya Bhasya - at least in the interpretation of Agathe Keller, who attributes her interpretation to a suggestion made by Takao Hayashi-contained the following diagram "to convince the dull-witted":92
Figure 3. There is a square corresponding to AC