________________
JOHANNES BRONKHORST
PĀŅINI AND EUCLID
value of 1. All that he demurs to is the claim that the result of this measurement in the case of a circle whose diameter is one - will be the square root of ten. This cannot be true, because the square root of ten, like other surds, cannot be expressed, presumably in terms of units or fractions
It is hard to know what "demonstration" would have convinced Bhaskara of the truth of the claim that = V10. But one thing is sure. The only other time he uses the term upaparti in this same commentary, it does not refer to anything like what we would call a proof. The passage concerned occurs in the middle of a discussion of an example under Ganitapada verse 6cd. Hayashi (1995: 75; modified) translates it as follows: "Drawing a plane figure in order to show the ground (upapatti) of the following Rule of Three".
Bhaskara goes on criticizing the claim that = 10 at great length. Part of his criticism is interesting because it reveals that he appears to have extended the respect which he felt for the rules contained in the Aryabhatiya to other rules, which he perhaps found in other treatises. One way in which he tries to demonstrate the insufficiency of this value for is by showing that it leads to totally unacceptable consequences The following passage illustrates this:64 And the calculation of an are on the basis of the assumption that the measurement of a circumference is made with the square root of ten, is not always (possible). For the stra for calculating an arc is the following] half arya verse:
The sum of a quarter of the chord and half the sagitta, multiplied by itself, ten times that, the square root of that.
Consider] now the following example: In (a circle) whose diameter is fifty-two, the length of the sagitta is two.
With the help of the rule "ogahnam vikthambham one obtains as length of the chord: twenty (20). With this chord the calculation of the arc becomes a quarter of the chord is 5, half the sagitta is 1, their sum is 6, multiplied by itself: 36, ten times that: 360, the square root of that is the arc.
The square of the whole chord is four hundred, the square of the are three hundred sixty, how is that possible? The are must certainly be longer than the chord. Here on the other hand the are being thought out by these extremely clever thinkers has turned out to be shorter than the chord! For this reason, homage be paid to this root of ten charming but not thought out.
Sanskrit; it is not known where it comes from, but clearly it offers a calculation involving the square root of ten. Bhaskara uses this rule. and shows that it leads to an absurd outcome. Strictly speaking this may be due (i) to the particular value assigned to T; (ii) to the form of the rule in general, quite apart from the value assigned to ; or (ii) to both of these at the same time. Bhaskara triumphantly concludes that the fault lies with the square root of ten. A small calculation would have shown him that the approximate value of which he, following Aryabhata's Ganitapada verse 10, does accept ( = 3.1416), if used with the same but adjusted formula, leads to the same absurdity (6 x 3.1416 = 18.8496 for the length of arc: 20 for the chord). We must conclude that - unless Bhaskara had lost his mathematician's mind while writing this passage - he really criticized the formula, which may have occurred in a presumably Jaina treatise.
This allows us to draw some further conclusions. Bhaskara might from a modern point of view, have criticized the wrong formula for calculating a length of arc by contrasting it with a correct one, along with a proof for the latter. He did not do so, most certainly because he did not have a correct formula, much less a proof for it. What is more, it is highly unlikely that he thought in terms of proofs of correct formulae. From his point of view the rules and theorems he had inherited were correct, the ones others had inherited were presumably mostly correct, but some of these led to unacceptable results, and were therefore incorrect.
The preceding reflections teach us the following. Judging by the evidence discussed so far, classical Indian geometry and grammar share a number of features, which are compatible with (but do not prove) the assumption that Panini's grammar did indeed exert an influence on the former. Two features in particular deserve mention:
This passage cites two rules, both of which Bhaskara calls sūtras. The second of these is "ogahunam vikkhambham", which had been cited in full, and illustrated, on the preceding page (p. 73 I. 2 ff.); it is in Prakrit and no doubt derives from a Jaina text. Contentwise this rule is no different from verse 17cd of the Ganitapada, studied above, and it is surprising that Bhaskara does not mind citing (and obviously accepts without questioning) the rule in the form in which it was used by the Jainas. The first sütra cited in this passage, on the other hand, is in
1. Classical Indian geometry, like grammar, describes objects that exist
in the material world, not abstractions. The practice of geometry does not therefore exclude the physical manipulation of such objects and the search for generalizations based on concrete measurements. This explains that some conclusions - such as the not totally spherical shape of a sphere - may be based on reflections about or
observations of material objects. 2. The objects of both classical Indian geometry and Sanskrit grammar
are described with the help of rules that are as general as possible. In the case of geometry, the resulting rules look like Euclidean theorems, but unlike the latter, and like the rules of Panini,
