Panini And Euclid Reflections On Indian Geometry

Author(s): Johannes Bronkhorst
Publisher: Johannes Bronkhorst
Catalog link: https://jainqq.org/explore/269453/1

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PANINI AND EUCLID: REFLECTIONS ON INDIAN GEOMETRY*
Professor Ingalls - in an article called "The comparison of Indian and Western philosophy" - made the following interesting observation (1954: 4): "In philosophizing the Greeks made as much use as possible of mathematics. The Indians, curiously, failed to do this, curiously because they were good mathematicians. Instead, they made as much. use as possible of grammatical theory and argument." This observation should not as goes without saying in our day and age be read as a description of the Indian "genius" as opposed to that of the Greeks (at least not in some absolute sense), but as a reminder of the important roles that mathematics and linguistics have played as methodical guidelines in the development of philosophy in Greece and in India respectively. Ingalls appears to have been the first to draw attention to this important distinction.
He was not the last. Ingalls's observation has been further elaborated by J.F. (= Frits) Staal in a few articles (1960, 1963, 1965).' Staal focuses the discussion on two historical persons in particular, Euclid and Panini, both of whom as he maintains have exerted an important, even formative, influence on developments in their respective cultures. Staal also broadens the horizon by drawing other areas than only philosophy into the picture. To cite his own words (1965: 114 = 1988: 158):
Historically speaking, Panini's method has occupied a place comparable to that held by Euclid's method in Western thought. Scientific developments have therefore taken different directions in India and in the West. Panini's system produced at an early date such logical distinctions as those between language and metalanguage, theorem and metatheorem, use and mention, which were discovered much later in Europe. In other Indian sciences, e.g, in mathematics and astronomy, as well as in later grammatical systems of Sanskrit, Prakrit, and Tamil, systematic abbreviations are used which not only are ingenuous but also constitute new adaptations of the same method.
The statement "Scientific developments have therefore taken different directions in India and in the West" is of particular interest. It suggests that there are methodical differences between Euclid's Elements and Panini's grammar. Staal does not tell us what they are. On the contrary, he emphasizes their common features, as in the following passage (1965: 113 1988: 157)
Journal of Indian Philosophy 29: 43-80, 2001.
©2001 Kluwer Academic Publishers. Printed in the Netherlands.

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When comparing Pänini's system with Euclid's Elements, a characteristic of the latter, i.e., deduction, appears absent from the former. It is true that there is a kind of deduction in Panini's grammar: dadhyatra is deduced from dadhi atra, and other forms are similarly deduced with the help of rules. But such deductions do not seem to attain the same degree of generality as Euclid's proofs. However, the difference reflects a distinction of objects, not of structure. In Euclid's geometry, propositions are derived from axioms with the help of logical rules which are accepted as true. In Panini's grammar, linguistic forms are derived from grammatical elements with the help of rules which were framed ad hoc (ie., sutras). Both systems exhibit a structure of logical deduction with the help of rules, and both scholars attempted to arrive at a structural description of facts. In both systems, contradictions and unnecessary complications are avoided. In both cases, the aim is adequate and simple. description.
In the immediately following paragraph Staal does mention that there are differences ("More detailed investigations into the methods of Euclid and Panini would throw light on points of difference as well"), but he does not say which ones. Instead he mentions one more common feature: "Another common characteristic is the above-mentioned desire to shorten principles (where Euclid pays attention to minimum number, Panini to minimum length), while disregarding the length of derivations." One looks in vain for a specification of the methodical differences between Euclid and Panini which might explain that "therefore" scientific developments have taken different directions in India and in the West.
Thirty years later Staal still emphasizes the importance of Euclid and Panini, and invokes them as typical examples of the scientific developments in their respective cultures. About Euclid he says (1993: 7): "The ancient Greeks developed logic and a notion of rationality as deduction best exhibited by Euclid's geometry. These discoveries contributed substantially to the development of Western science." About India he observes (1993: 22): "Ancient Indian civilization was an oral tradition and the oral transmission of the tradition became the first object of scientific inquiry. Thus arose two human sciences, closely related to each other in their formal structure: the sciences of ritual and language." The science of ritual, he points out, formulated ordered rules which express the regularities of Vedic ritual. He then continues (pp. 23-24): "The Sanskrit grammarians used rules of precisely this form and demonstrated in a similar manner why many of them have to be ordered. In consequence of metarules, rule-order and other formal properties of rules, Pänini developed Sanskrit grammar as a derivational system in some respects more sophisticated than the deductive system of Euclid..." But when one searches in this later publication for a specification of the methodical differences between Pänini and Euclid, one is once again disappointed. Here, too, one finds rather an
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emphatic statement to the effect that the two are essentially similar (1993: 24):
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Science is universal and its main branches developed in all the great civilizations. ...The great traditions of Eurasia are basically similar but there are differences between them in character and emphasis. West Eurasian science is characterized by an emphasis on nature and punctuated increases and decreases in theoretical and empirical sophistication;... Indian science by formal analysis and an emphasis on human theory.
The "therefore" of "Scientific developments have therefore taken different directions in India and in the West" still remains shrouded in mystery.
The first question that has to be addressed at this point is the following: Is it true that scientific developments have taken different directions in India and in the West? Recall that Ingalls, in the passage cited above. observed that "the Indians, curiously, failed to [make as much use as possible of mathematics in philosophizing], curiously because they were good mathematicians." Indeed, the mathematical literature preserved in India is extensive, though little studied;2 M.D. Srinivas gives the following estimate (1990: 30): "The recently published 'Source Book of Indian Astronomy13 lists about 285 works published in mathematics and mathematical astronomy, of which about 50 are works written prior to the 12th century A.D., about 75 are works written during 12-15 centuries and about 165 are works written during 16-19th centuries." A calculation on the basis of all the volumes of David Pingree's Census of the Exact Sciences in Sanskrit (including the one that was not yet accessible to the authors of the 'Source Book of Indian Astronomy") might reveal even larger numbers of works than this.4 Mathematics, astronomy and medicines were clearly not neglected in classical India. We shall see below that geometry, in any case, was not neglected in Vedic India either.
Should we then drop the thesis of the relative importance of geometry and grammar in Greece and India respectively? Did Pänini not exert the influence that Ingalls and Staal attribute to him? Rather than drawing any such abrupt conclusion at his point, I would suggest that the thesis of Ingalls and Staal remains interesting, even though it needs, and deserves, closer attention, and may have to be tested against more evidence than it has been exposed to so far. It is an undeniable fact that many, if not all, classical Sanskrit authors had studied Sanskrit grammar and therefore undergone the influence of Panini and his early commentators. The extent to which this omnipresent influence affected philosophy and the sciences remains to be determined, but there is no

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dispute over the fact that such an influence could have taken place, if for no other reason than that the authors concerned had spent time studying grammar.
One rather obvious question has rarely been asked, certainly not by Ingalls and Staal, viz. to what extent grammatical influence is noticeable in Indian mathematical literature, and the extent to which Panini's work may have provided a methodical guideline for the authors of mathematical texts. This question will gain in importance if we concentrate on the portions on geometry in Indian mathematical texts, because here we will be able to make a direct comparison with Euclid. If we find obvious differences in method between Euclid and classical Indian geometry, we may then ask whether and to what extent these differences are due to the influence of Panini.? The fact that the geometry of Indian mathematical astronomy may ultimately derive from Euclidean geometry will make this question all the more poignant.
Let me cite a second time the following passage from Staal's article "Euclid and Panini": "When comparing Panini's system with Euclid's Elements, a characteristic of the latter, i.e., deduction, appears absent from the former. It is true that there is a kind of deduction in Panini's grammar: dadhyatra is deduced from dadhi atra, and other forms are similarly deduced with the help of rules. But such deductions do not seem to attain the same degree of generality as Euclid's proofs. However, the difference reflects a distinction of objects, not of structure." This statement claims that the fact that deductions of a certain type are present in Euclid and absent in Panini is due to the distinction between the objects which these two sciences are dealing with. It strongly suggests that a competent Indian dealing with geometry rather than with grammar would use the same kind of deductions as Euclid.
.46
This, of course, is a testable suggestion, and the remainder of this article is meant to test it against the evidence provided by Bhaskara's commentary on Aryabhata's Aryabhatiya, one of the earliest texts that provides us first-hand information about classical Indian geometry in practice. Before however doing so, it will be useful to specify somewhat more precisely what the methodical influence of Panini's would look like. What are the principal characteristics of Panini's. grammar?
This question was already raised in the Mahabhāṣya of Patanjali, a grammatical work dating from the second century B.C.E. whose study accompanied that of Panini's grammar throughout most of the classical period. That is to say, students who had to acquaint themselves with Panini's grammar also had to learn the analysis of its method that is
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presented in the Mahabhäṣya. This analysis occurs in the introductory chapter of the Mahabhāṣya (the Paspaśähnika), no doubt the best known portion of this work. Starting from the position that grammar should teach words, Patanjali then raises the question as to how this is to be done. There are far too many of them to be enumerated one by one. Patanjali then continues:10 Words must be learned by means of general principles and specific features, laid down in general rules and exceptions. This is indeed how Panini's grammar works: rules are formulated that are as general as is possible; where necessary exceptions are added to make sure that the general rule will not be applied where this is not desired. The greatest possible generality, in combination with extreme concision of expression, constitute the two most important characteristics of Panini's grammar.
With this in mind we can turn to the mathematical texts. The Aryabhatiya of Aryabhata (= Aryabhata I) dates from 499 (Pingree) or 510 (Billard) C.E., and contains a chapter on mathematics entitled Ganitapada which is "the earliest text of this genre that we have" (Pingree, 1981: 56). The Aryabhatiya is written in äryä verses which mainly contain rules formulated in a highly condensed form. The rules of the Ganitapada in part concern arithmetic, in part geometry. Indeed. the commentator Bhaskara (to be introduced below) clearly states that mathematics (ganita) is twofold: arithmetic and geometry, or perhaps more precisely: arithmetic of quantities and arithmetic of geometrical figures, 12
From among Aryabhata's rules, let us concentrate on the probably best known theorem of Euclidean geometry, viz. the Pythagorean theorem. This theorem finds expression in the first half of verse 17 of the Ganitapada, which reads:13 "The square of the base [of a rightangled trilateral] and the square of [its] upright side is the square of the hypotenuse." No more is said about it: no proof is given, no examples and no diagrams. We have seen that Staal looks upon the concision of Indian mathematical texts as an adaptation of Panini's method. Others maintain that this conciseness is a sign of the scientific character of Aryabhata's text, as does Pierre-Sylvain Filliozat in the following passage:14
Cette forme littéraire, en particulier le recours à l'ellipse de l'expression et de la pensée, a une autre raison d'être, en plus de l'obéissance à la convention culturelle d'un pays et d'une époque. Elle a une utilité de caractère scientifique, en ce qu'elle relève de la formalisation. Les intellectuels indiens ont reconnu très tôt l'aide apportée à la pensée par l'économie de l'expression, l'outil efficace qu'est un langage technique abrégé. Le premier exemple de formalisation scientifique dans l'histoire universelle des sciences semble bien être le formulaire de linguistique qu'est le sutra de Panini et qui se caractérise par l'emploi d'une métalangue et le passage obligé au niveau de

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la plus grande généralité afin de permettre l'application d'une formule unique au plus grand nombre possible de cas particuliers. Aryabhala n'a pas poussé la formalisation aussi loin que le grammairien. Mais il ne l'a pas ignorée. Il n'a pas de métalangue à proprement parler, mais il a son vocabulaire technique et surtout il vise le plus haut degré de généralité dans ses propositions. Ceci est une des valeurs scientifiques de son oeuvre
Filliozat is right in that the Pythagorean theorem is presented in its most general form. Instead of enumerating a number of values valid for specific rectangular triangles (3: 4: 5; 5: 12: 13; etc.) Aryabhata formulates this rule for all possible rectangular triangles, and for all possible values of its sides. One might consider the possibility that the very (re)discovery of, and interest for, the Pythagorean theorem (and of other theorems) by Indian mathematicians is due to the obligation to formulate their knowledge in as concise, and therefore as general, a form as possible. But it must be added that the Ganitapada, perhaps because of this same "Paņinian" conciseness, presents the Pythagorean theorem (and many other theorems, as we will see) without a proof, and therefore without a hint as to why we should accept it. One might be tempted to conclude from this that the search for brevity, which tumed out to be such a great advantage for grammar, was a mixed blessing for mathematics. We will see below that such a conclusion would not do full justice to all the available evidence.
The part of Brahmagupta's Brahmasphutasiddhanta (628) dedicated to geometry (12.21-47) is as concise as Aryabhata's text, and much the same could be said about it. The Pythagorean theorem is formulated as follows: Subtracting the square of the upright from the square of the diagonal, the square-root of the remainder is the side; or subtracting the square of the side, the root of the remainder is the upright: the root of the sum of the squares of the upright and side is the diagonal." Here too, no proof, no examples, no diagrams.
A third text on mathematics which may perhaps date from a period close to Brahmagupta (Hayashi, 1995: 149), the Bakhshäli Manuscript contains in its preserved portions little about geometry (see Hayashi, 1995: 97): essentially only the sutra which Hayashi calls N14 (p. 401: Sanskrit pp. 221 ff.: English translation pp. 322 ff.). This sutra which prescribes a way to calculate the volume of a particular solid, is accompanied by an example. The fragmentary state of this text does not allow us to draw far-reaching conclusions.)
It is not altogether fair to judge highly condensed fundamental texts from classical India without taking into consideration that such texts were always accompanied by one or more commentaries, whether oral or written. This is true for Panini's Astadhyayi, which was the
starting point for an extensive commentatorial tradition, and which must have been accompanied by some kind of commentary from the very beginning. Patanjali's Mahābhāsya - which is a commentary on the Astädhyayi, even though a special kind of commentary - was composed well before the beginning of the common era. More straightforward commentaries on the Astādhyayi no doubt existed from an early time onward, and even if the oldest surviving commentary of this nature - the Kasika Vrtti - is a relatively late text (around 700 C.E.), it is known that earlier such commentaries existed." Aryabhata's text, too, may initially have been studied under the guidance of a teacher, who would orally provide many of the elements that are lacking in the verses. Written commentaries were subsequently composed, the earliest ones of which have not survived. The earliest mathematical commentary that has survived is the Aryabhatiya Bhäsyal of Bhaskara (= Bhaskara I), which dates from 629 C.E.,20 i.e. almost exactly from the same year as Brahmagupta's Brahmasphutasiddhanta. By contrast, the earliest surviving commentary on Brahmagupta's Brahmasphutasiddhanta - Prthudakasvamin's Vivarana - dates from 864, and belongs to the later period of Sridhara and others which will not be considered here.
It is in Bhaskara's Aryabhatiya Bhasya, then, that we will look for the elements that are so obviously missing from Aryabhata's Ganitapada - assuming, for the sake of argument, that Staal's assessment of Indian geometry is correct.21 Three of these elements have already been mentioned. What is primarily missing in the fundamental text are a proof of the theorem concerned, examples of how and where it can be applied, and where necessary one or more geometrical diagrams that facilitate visualizing the theorem and its proof; in view of the Indian commentatorial customs in other areas we may also expect to find explanations and (re-)interpretations of the exact formulation of the basic text, where necessary.
With this fourfold expectation in mind we turn to the Aryabhatiya Bhasya on Ganitapada 17ab. We do indeed find here an explanation of the words of the basic text (which in this case is very short, given the unproblematic formulation of the theorem)22 followed by examples (tasks with solutions, all of which concern triangles the lengths ol whose sides relate to each other as 3: 4: 5) and diagrams. The only thing that is lacking is a proof of the theorem!
For a reader schooled in Euclidean geometry this is very surprising. An important, nay fundamental, theorem of geometry is presented. explained and illustrated by Bhaskara without proof:23 This absence

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of proof does not stop Bhaskara from frequently using this theorem elsewhere in his commentary. He cites it in the form given to it by Aryabhata in verse 17ab in his comments on Ganitapada 6ab (p. 56 1. 5), 6cd (p. 58 II. 14-15), 17cd (p. 103 II. 12-13), and uses it implicitly at many other occasions. The theorem is already implied in the explanation-cum-etymology which Bhaskara gives of the difficult term karani "surd" in his introductory remarks to verse 1:24 "The karani is [so called] because it makes (karoti) the equation of the diagonal (c) and the sides (a and b) [of a rectangle] (a2+ b2 = c2)."
It goes without saying that the absence of a proof in the case of the Pythagorean theorem raises the question whether any proofs are presented in connection with other theorems. This question has been investigated by Takao Hayashi in the Introduction of his book The Bakhshali Manuscript (1995), where he studies in particular the use of the terms pratyaya(-karana) and upapatti. We will consider some of the passages concerned below, but note here already that Hayashi arrives at the following conclusion (p. 76): "As a matter of fact... we can hardly find proofs or derivations of mathematical rules in Bhaskara's commentary on the Aryabhatiya."
To illustrate this point, we shall consider a passage which might at first sight create the impression that it does provide - or rather: hint at - a proof of a rule; we will see that it doesn't. The rule concerned is presented in the second half of verse 17: "In a circle the product of the two sagittas of the two arcs [that together constitute the circle] equals the square of half the chord."25
B
E
Figure 1. CD x DE = AD2.
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That is to say: CD x DE = AD2. This rule can easily be proved with the help of the Pythagorean theorem:
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AC2 = AD2 + CD2 AE2 = AD2 + DE2
CE2 (CD + DE)2= AC2+ AE2 = AD2+ CD2 + AD2 + DE2 CD2+ DE2 + 2 x CD x DE = CD2+ DE2 + 2 x AD2 CD x DE = AD2 q.e.d.
Bhaskara does not however do so, at least not at the beginning of his comments on this rule. Instead he gives a large variety of examples (tasks with solutions), which cover more than six pages of the edition. Having provided (numerical) solutions to all the problems Bhaskara then states:26 The verification (pratyayakarana) in the case of all these surfaces [is obtained] with the [rule according to which] 'the square of the hypotenuse equals the sum of the squares of the two rectangular sides'." In spite of the impression which this statement might create, this is not a reference to the proof of the theorem, but a reminder that the Pythagorean theorem can be used, instead of the present theorem, to carry out the calculations and find the solutions to the problems discussed. That is to say, the Pythagorean theorem can be used to verify the solutions that had been reached with the help of the other theorem. This may be a justification of sorts of that theorem, but hardly a proof in any of its usual senses.27
If we follow Staal in thinking that the study of geometry should more or less automatically lead to deductions of the Euclidean kind, then the absence of proofs in Bhaskara's commentary demands an explanation. It might then be symptomatic of a fundamental difference between Indian and Greek geometry. This difference in its turn, still according to Staal, might be due to the methodic influences of Panini and Euclid respectively. Is it conceivable that Panini's grammar may have had this effect on classical Indian geometry? Let us consider some aspects of this question in more detail.28
Panini's grammar, as was pointed out above, had been studied by all or almost all classical Brahmanical scholars in whatever disciplines, as well as by others (such as Buddhist and Jaina scholars). It is almost superfluous to recall that Bhaskara's commentary contains ample signs that its author, too, was thoroughly acquainted with it (even though his text, in its edited form, contains a number of grammatical infelicities; see note 59 below). The fourth appendix to K.S. Shukla's edition of the text enumerates eight quotations from Panini's Astādhyāyī, one from its

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Dhatupatha, ten from Patanjali's Mahabhäsya,29 one from Bhartṛhari's Vakyapadiya,30 and one passage referring to the list of sounds that precedes Panini's grammar (the "Sivasütras"). This enumeration is however far from complete.31 A quotation from the Mahabhāṣya that has not been identified by the editor is the line samanyacodanās ca visese 'vatisthante (p. 39 1. 23; p. 55 11. 12-13), which is cited from the Mahabhasya on Pän. 4.1.92 vt. 3.32 Other quotations that could be added to the list are the following. The comparison yatha 'kniti ca' ity atra luptanirdisto gakāraḥ (p. 9 11. 18-19) refers, without saying so, to a discussion in the Mahabhäṣya on Pän. 3.2.139. The line yas ca sarvas ciram jivati sa varsaśatam jivati (p. 13 1. 16) echoes a line from the Paspaśähnika;33 drstänuvidhitväc chandasaḥ (p. 14 1. 22) is part of Pan. 1.1.6 vt. 1; ingitena cestitena nimișitena mahatä vä sütraprabandhena [ca] äcäryäṇām abhipräyo gamyate (p. 34 11. 16-17) is quoted from the Mahabhäṣya on Pän. 6.1.37:34 tätsthyät täcchabdyam (not chabdyam; p. 35 l. 10) occurs on Pän. 5.4.50 vt. 3.35 Particularly interesting is the line anityaḥ samäsäntavidhiḥ (p. 23 1. 22); this reflects a position taken in the Mahabhäṣya on Pän. 6.2.197 (III p. 140 1. 6: vibhāṣā samäsänto bhavati) but is identical in form with a grammatical paribhāṣā which we find, perhaps for the first time, in Vyadi's Paribhāṣāvṛtti (no. 69: samäsäntavidhir anityah: Wujastyk, 1993: I: 70).36 Equally interesting is Bhaskara's use of the grammatical expression sambandhalakṣaṇā ṣasthi (p. 38 1. 2), which does not appear to occur in the Mahabhäṣya, but which is attested in the Käsikävṛtti (on Pän. 7.1.90). The editor has furthermore not identified a i un rlk (p. 181. 16), which is "Śivasutra" 1 & 2. To this must be added that there are many implied references to sutras of Panini. One occurs on p. 60 11. 8-9 of the edition, where the correct reading must be anyapadarthena (rather than anyapädärthena) samaparinähaśabdena kṣeträbhidhänät, which means: "because the expression samaparinäha - [being a bahuvrihi compound and therefore] referring to something else (on account of Pän. 2.2.24: anekam anyapadarthe (bahuvrihih)] - would denote the field". Similarly, p. 116 11. 2-3: trirašiḥ prayojanam asya ganitasyeti trairäsikaḥ "the trairäsika ('rule of three') is called thus because three quantities (triräsi) are the purpose (prayojana) of this calculation" contains an implicit reference to Pän. 5.1.109 prayojanam, which prescribes the suffix thaN (= ika). The immediately preceding line. trayo rasayaḥ samährtäḥ trisäsih "three quantities combined are [called] trirasi implicitly refers to Pan. 2:1.51 (taddhitärthottarapadasamahāre ca) and 52 (sankhyapurvo dviguḥ) which prescribe the compound called⚫ samahāra dvigu. which is a kind of tatpurusa. P. 7 1. 3 tasmin krakace
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bhavaḥ kräkacikaḥ and p. 111. 3 navante bhavam navantyam similarly refer to Pāṇ. 4.3.53 tatra bhavaḥ, p. 180 1. 15 adau bhavati iti adyaḥ to Pan. 4.3.54 (digädibhyo yat). P. 2 1. 4 ktväpratyayena purvakälakriya 'bhidhiyate evokes Pān. 3.4.21 samanakartṛkayoḥ purvakäle (ktvä 18). In a way each and every analysis of a compound of which there are numerous instances in the Aryabhatiya Bhasya, even though their technical names are rarely used - can be considered an implicit reference to Panini's grammar. The influence of Patanjali's Mahabhāṣya on early mathematical literature can also be deduced from the fact that Bhaskara refers to the works of Maskari(n), Pürana and Mudgala as providing the lakṣaṇa and the lakṣya of various branches of mathematics; these are precisely the terms used in the Paspasähnika of the Mahabhāṣya to designate grammar: lakṣana designates the sutras, lakṣya the objects to be studied, i.e. words in the case of grammar." It is in this context interesting to note that Bhaskara refers to Aryabhata's text using these same expressions lakṣaṇa and sutra: lakṣaṇa infrequently, e.g. p. 74 1. 3; sutra very often.38 This means that Bhaskara, like Patanjali, uses the terms lakṣaṇa and sutra as synonyms. This is particularly clear on p. 49 1. 16, where the two are simply juxtaposed to each other: p. 51 1. 12 cites a lakṣaṇasutra. [It is clear from his discussion that these sūtras do not always coincide with the verses of Aryabhata,39 but can be parts of them. An extreme case is verse 19 of the Ganitapada, in which Bhaskara distinguishes no fewer than five different sutras."]
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The presentation of geometrical theorems without proof raises the question whether perhaps proofs were known to the authors concerned, but were not recorded in their texts because their literary model - Panini's grammar and its commentaries, as well as other texts that had adopted this model - did not really leave place for them. It is at least conceivable that Aryabhata and Bhaskara (or the mathematicians from whom they took their material) practiced geometry on a more abstract level than is evident from their writings, making sure that the theorems they presented to their readers were based on solid proofs." One would be free to think so, were it not that these two authors present some theorems that are quite simply wrong.42 This cannot but mean that these theorems were not based on solid proofs, for no such proofs are possible. (It does not exclude the possibility that some form of verification existed; this system of verification, if it existed, was not however capable of discovering and correcting these errors.)
Aryabhata is wrong where he gives the volume of a pyramid as:43 "Half the product of the height and the [surface of the triangular base]

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is the volume called "pyramid"." The correct volume of a pyramid is a third, not half, of the product here specified. In spite of this, Bhaskara accepts Aryabhata's rule and carries out some incorrect) calculations with its help. The same is true of Aryabhata's incorrect rule for the volume of a sphere 44
Aryabhata's incorrect rules have drawn the attention of scholars, some of whom have tried to interpret the rules differently, so as to obtain a correct result.45 Filliozat and Mazars object to such a reinterpretation in a joint publication (1985), pointing out that Bhaskara's comments do not support it. They further make the following important observation (p. 40): "Il faut comprendre Aryabhata, non pas dans ses seuls résultats transposés en formules modernes, mais dans sa propre culture intellectuelle. In a more recent publication, Filliozat states (1995: 39): "It is an error to actualise the ideas of past scientists, to transpose in modem terminology their ancient expressions."
Filliozat and Mazars are no doubt right in this, and one would expect that they might try to explain Aryabhata's errors in the light of "sa propre culture intellectuelle". Unfortunately they don't do so. These crrors remain a total mystery to them, as is clear from the concluding words of their article (p. 46):
chapter, the Kālakriyapada, Bhaskara cites, under verse 5. a verse from an earlier (non-identified) astronomical text-about the months in which the sun passes the summer and winter solstices - and calls it Smrti, - more precisely: our Smrti (p. 182 1. 9: asmäkam smrtih); he bolsters this position with an argument based on Mimamsă, unfortunately not completely clear in all its details, but which in outline looks as follows: this Smrti has to be accepted, for it is neither in conflict with the Sruti. nor with perception. Calling a text Smrti and insisting that it conflicts neither with the Sruti nor with perception amounts to granting it the same authority as the Veda itself, and maintaining that it cannot be wrong.
It seems clear from what precedes that there is little space in Bhaskara's geometry for proofs, and none at all for definitions and axioms, the starting points of perfect proofs. This raises the following fundamental question. If the geometrical figures dealt with by the Indian mathematicians under consideration are not constituted of elements laid down in definitions So what then is the geometry of Aryabhata and Bhaskara about? If its triangles, circles, pyramids and spheres are not abstractions, what then are they?
We may find a clue to the answer in the following passage of Bhaskara's Bhäsya on Ganitapada verse 11:3
L'existence de telles erreurs chez le grand mathématicien est certes surprenante. Elle l'est encore plus chez Bhaskara, puisque nous savons que son contemporain Brahmagupta connaissait, au moins pour le volume de la pyramide, la formule exacte
We, on the other hand, maintain that there is a chord equal to the arc (which it subtends). If there were no chord equal to the arc fit subtends, an iron ball would not be stable on level ground. We infer from this that there is an area such that the iron ball rests by means of it on the level ground. And that area is the ninety-sixth part of the circumference.
Other teachers, too, have accepted u chord equal to the arc (which it subtends as is clear from the following quotation: "From the body of a sphere. a hundredth part of its circumference touches the earth.52
And yet it seems obvious that these mistakes do throw light on the intellectual culture of their authors. It seems inevitable to conclude that the theorems propounded by Aryabhata and Bhaskara were apparently not accompanied by proofs, not even in private, not even outside the realm of the written commentary. They were handed down as received truths, with the result that incorrect theorems were not identified as a matter of routine by any student who checked the proofs." Bhaskara confirms that he regards the theorems and other information contained in Aryabhata's work as received truths in his commentary on the first chapter, the Gitikāpāda. Here he states, under verse 2, that all knowledge derives from Brahma: Aryabhata pleased Brahmä on account of his great ascetic practices and could then compose, for the well-being of the world, the ten Gitikasutras on planetary movement (chapter 1), as well as the one hundred and eight arya verses on arithmetic (ch. 2). time (ch. 3) and the celestial globe (ch. 4)." Elsewhere (p. 189 11. 14-15) he calls Aryabhata atindriyarthadarsin "seeing things that are beyond the reach of the senses". It is further noteworthy that in the third
Both Bhaskara's own words and the line he cites seem to confuse the surface of a sphere and the circumference of a circle. This may for the moment be looked upon as a detail. The most important thing to be learnt from this passage is that for Bhaskara - and no doubt for other teachers as well - spheres and circles are no (or not only) abstract objects, but also) concrete things whose features are at least to some extent determined by their behavior in the world of our daily experience. A sphere, we learn, has to have flat surfaces, for only thus can it be stable on level ground.
The question of the absence of proofs in Indian geometry acquires a different dimension once it is clear that Euclidean and classical Indian geometry may not really concern the same objects. The objects of classical Indian geometry - its triangles, circles, spheres etc. -

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Figure 2. The area of triangle ABC is half the area of rectangle EFCB.
A passage in Bhaskara's comments on Ganitapada verse 10 is interesting in this context. We read here:59
are no mere abstractions, but things present in the outside world. (The fact that the use of triangles formed by such imprecise things as shadows and flames inevitably implies that the mathematician worked with idealized objects, does not necessarily contradict this.) As such they resemble the objects of grammatical analysis far more than do Euclidean diagrams. The objects of grammar - sounds, words, phrases - are there in the outside world, and grammar is therefore a science which deals with objects whose existence is quite independent of the volition of the grammarian. To invoke Patanjali's words: one does not visit a grammarian when in need of words as one visits a potter when one needs a jar. In other words, the objects of grammar, unlike pots, are not fashioned in accordance with the wishes of their users. Elsewhere Patanjali states: "Words are authoritative for us; what the word says is our authority." Once again, words do not adjust to grammarians, but grammarians follow the dictates of words. Grammar presents multiple linguistic data with the help of rules that are as general as possible, but that cannot go against the data of language. Similarly, the rules of classical Indian geometry describe numerous geometrical forms that exist in the outside world with the help of rules that are as general as possible. The fact that these rules describe concrete objects rather than mathematical abstractions has as consequence that they may often be generalizations from concrete observations, rather than general statements derived from first principles. The Pythagorean theorem, seen this way, may therefore in the classical Indian context be a generalization of measurements taken from various rectangular triangles; the notion of a general proof for this theorem might have appeared to the early Indian mathematicians as completely out of place, just as to a grammarian a general proof of the grammatical rule that i followed by a is replaced by y, not only in dadhyatra out of dadhi atra but also in all other similar situations, would have looked strange
Having said this, it should not of course be overlooked that geometrical figures and bodies are not the same kinds of things as linguistic expressions. Numerous geometrical rules become obvious with the help of minimal manipulation of figures. Take the rule that the area of a triangle is half the product of its height and its base." This rule follows almost automatically from a look at the following diagram, and there is no need to assume that it is a generalization of a number . of measurements 5* And yet the notion of general proof does not characterize Bhaskara's discussions of geometry.
[Question:) Why is the approximate circumference for a circle given, nor its exact circumference? (Response:) People think as follows: there is no way in which the exact circumference (can] be calculated [Question:] Isn't there the following saying the circumference of a circle is the square root of ten times the square of the diameter [Response) Here too, the claim that the circumference of a diameter that equals one is the square root of ten is mere tradition, and no demonstration (papani). [Objection:) And if one thinks that the circumference, as directly prutyakena) measured, of a circular) area whose diameter equals one is the square root of ten, [Response) this is not correct), for the measure of surds cannot be expressed (Objection:) If it is maintained that the circumference of a circle] having that diameter (viz one), surrounded by the diagonal of an oblong quadrilateral whose width and length are one and three frespectivelyl. has the same size (as that diagonal). Response:) that, too, has to be established.
This passage is interesting for various reasons. For one thing, it uses the expression upaparti, which some translate "proof", but for which "demonstration" seems more appropriate. It also states quite unambiguously that "the measure of surds cannot be expressed in fractions (7)". But its special interest for us at present lies in the fact that it mentions, and accepts in principle, the idea of direct measurement as a method to reach conclusions of a geometrical nature. It even indicates how the square root of ten can be measured, so as to compare its length with the circumference of a circle. It appears to be legitimate for Bhaskara to measure the circumference of a circle, presumably with a cord, and derive from this measurement what we would call the

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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,

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they do not derive their validity from proofs. This explains that some incorrect rules have been able to slip into the works of Aryabhata and Bhaskara and remain undiscovered for a long time.
The notion of proof that is claimed to have existed in India and elsewhere by the above-mentioned authors has been examined by G.E.R. Lloyd in the 3rd chapter of his book Demystifying Mentalities (1990), some passages of which are worth quoting. Lloyd begins as follows (p. 74): At the outset we must be clear that "proof' and 'proving' may signify a variety of more or less formal, more or less rigorous, procedures. In some domains, such as law, proving a fact or a point of law will be a matter of what convinces an audience as being beyond reasonable doubt. Again in some contexts, including in mathematics, "proving a result or a procedure will sometimes consist simply in testing and checking that it is correct. Both of these are quite informal operations. But to give a formal proof of a theorem or proposition requires at the very least that the procedure used be exact and of general validity, establishing by way of a general, deductive justification the truth of the theorem or proposition concerned. More strictly still Aristotle was to express the view that demonstration in the fullest sense depended not just on deductive (he thought specifically syllogistic) argument but also on clearly identified premises that themselves had to fulfil rather stringent conditions ... He was the first not just in Greece, but so far as we know anywhere. explicitly to define strict demonstration in that way
Two crucial distinctions have, then, to be observed, (1) between formal proofs and informal ones, and (2) between the practice of proof (of whatever kind) and having an explicit concept corresponding to that practice, a concept that incorporates the conditions that need to be met for a proof to have been given.
Before we jump to the conclusion that these shared features are due to the influence of Paninian grammar on classical Indian geometry, some further facts and arguments have to be taken into consideration. The claimed absence of proofs in classical Indian geometry, in particular, has to be confronted with the conflicting claim that proofs existed already in Indian geometry long before Aryabhata and Bhaskara, and presumably before Panini, viz., in the geometry of the Vedic Sulba Sütras. Frits Staal - whose views about the importance of grammar are central to this essay - maintains this position in a very recent article (1999)," but does not provide arguments to bolster it beyond referring to a number of publications by Seidenberg (1978, 1983; one might add 1962, 1975: 289 ff.) and Van der Waerden (1983: 26 ff.; one might add 1980)." Seidenberg and Van der Waerden - like Staal himself - argue for a common origin of mathematics, or of geometry specifically, as found in various cultures. Staal, for example, pleads for a common origin of Greek and Vedic geometry in Bactria/Margiana; Van der Waerden proposes "a Neolithic Geometry and Algebra, invented somewhere in Central Europe between (say) 3500 and 2500 B.C., in which the 'Theorem of Pythagoras' played a central role" (1980: 29; cp. 1983: 33–35); Seidenberg claims to prove that "(a) common source for the Pythagorean and Vedic mathematics is to be sought either in the Vedic mathematics or in an older mathematics very much like it" (1978: 329; similarly already Schroeder, 1884). More important for our present purposes is that these three authors -as pointed out aboveagree that Vedic mathematics had proofs. Van der Waerden goes further and maintains that already the original mathematicians (whom he calls the pre-Babylonian mathematicians) had them (1980: 8): "I suppose that these mathematicians had proofs, or at least plausible derivations. A pupil who has to solve a mathematical problem can do it just by applying a rule he has leamt, but the man who invented the rule must have had some sort of derivation. I also suppose that our pre-Babylonian mathematicians had a proof of the 'Theorem of Pythagoras'." Seidenberg is hardly less brazen (1978: 332): "The striking thing (in the Apastamba Sulvasutra) is that we have a proof. One will look in vain for such things in Old-Babylonia. The Old-Babylonians, or their predecessors, must have had proofs of their formulae, but one does not find them in Old-Babylonia."
Lloyd points out that the second distinction, in his view, has been ignored or badly underplayed in recent attempts to see the notion of proof as originating long before even the earliest extant Egyptian and Babylonian mathematics. He then turns to Vedic mathematics and argues that the authors of its key texts were not concerned with proving their results at, all, but merely with the concrete problems of altar construction. Vedic mathematics is again dealt with in a "Supplementary note: geometry and 'proof' in Vedic ritual" (pp. 98-104), where Lloyd observes that "the notion that the authors in question had a clear and explicit concept of proof is subject to the general doubt... that to obtain results is one thing, to have that concept as an explicit one is another. ... It also falls foul of one further fundamental difficulty. This is that no clear distinction is drawn in these texts between the rules that are expressed to arrive at what we should call approximations and those that are employed to yield what we should call exact results" (p. 101).
The question could be asked whether the notion of proof is really culture-specific to ancient Greece. Here Lloyd comments (p. 75): The practice of proof, in Greece, antedates by several generations the first explicit formal definition (first given by Aristotle in the fourth century B.C.) and the process whereby such notions us thul of the starting points or axioms came to be clarified wus both hesitant and gradual. That long and complex development, in Greece, belongs to and is a further instance of the gradual heightening of self-consciousness we have

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62
cxcmplified before, when Necond-order questions came to be raised concerning the nature, status, methods and foundations of different types of inquiry. None of the attendant circumstances surrounding these developments, and none of the steps by which the various interrelated key notions came to be made explicit, can be paralleled in Vedic literature or in the evidence for Vedic society
Does this mean that we should not expect any such notion to have existed in classical India? Is the notion of proof really a Greek invention, determined by the specific social and political situation prevailing in that culture? And does it follow that all cultures that do possess the notion of proof must have borrowed it - directly or indirectly from ancient Greece?
Before trying to answer these questions it seems appropriate to recall that the absence of explicit proofs and of an identifiable notion of proof seems to be a common feature of many cultures that produced geometry. 0. Neugebauer, for example, states about ancient Babylonia (1957: 4546; cited in Seidenberg, 1975: 286): "It must... be underlined that we have not the faintest idea about anything amounting to a "proof conceming the relations between geometrical magnitudes. Richard J. Gillings - in an appendix meant to counter some of the negative criticism addressed at the mathematics of the ancient Egyptians on account of its lack of formal proof - concludes nonetheless (1972: 234); "We have to accept the circumstance that the Egyptians did not think and reason as the Greeks did. If they found some exact method (however they may have discovered it), they did not ask themselves why it worked. They did not seek to establish its universal truth by an a priori symbolic argument that would show clearly and logically their thought processes." Chinese geometry, it appears, did not use proofs either. When early in the seventeenth century Euclid's Elements were translated into Chinese," by Matteo Ricci and Xu Guangqi, the latter of these two wrote in the preface to another work) that only after the translation of Euclid's Elements into Chinese had it become possible to transmit proofs and explanations. In fact, he maintained, the Westem methods of conveying are not essentially different from the methods transmitted in earlier Chinese treatises. What makes Western mathematics more valuable is that it supplies explanations which show why the methods are correct. Joseph Needham (1959: 91) confirms this by stating: "In China there never developed a theoretical geometry independent of quantitative magnitude and relying for its proofs purely on axioms and postulates accepted as the basis of discussion." We learn from Jean-Claude Martzloff's A History of Chinese Mathematics that in the Chinese tradition of geometry "the figures essentially refer not to idealities but to material objects which, when manipulated in
an appropriate manner, effectively or in imagination, may be used to make certain mathematical properties visible". This has various consequences, among them the use of empirical methods: "To show that the side of a regular hexagon inscribed in a circle has the same length as the radius, six small equilateral triangles are assembled and the result is determined de visu. One proof technique for determination of the volume of the sphere involves weighing it. Sometimes the reader is asked to put together jigsaw pieces, to look at a figure or to undertake calculations which themselves constitute the sole justification of the matter in hand". Martzlott concludes (p. 72). "If one has to speak of proofs', it might be said that, from this point of view, the whole of the mathematician's art consists of making visible those mathematical phenomena which are apparent not in Platonic essences but in tangible things". The concern with mathematical objects as parts of objective reality reminds us of the similar concern on the part of Bhaskara, studied above.
Should we then conclude that the notion of proof only belongs to ancient Greece and its inheritors? I think the situation is more complex than this. Recall that India at the time of Aryabhata and Bhaskara did have a clear notion of proof. Such a notion was present in Indian logic, which had been developing since long before Aryabhata and reached some of its peaks in the persons of Dignaga and Dharmakirti precisely during the period that separates Aryabhata from his commentator Bhaskara at any rate, it would appear, had no excuse for not being aware of the notion of proof, and for not providing proofs for his theorems. At first sight the situation of these two mathematicians should not therefore be very different from Euclid in Greece Greek philosophers developed the notion of proof in their logic, and Greek mathematicians did the same in geometry. Indian philosophers developed the notion of proof in their logic, but the Indian mathematicians did not follow suit. Why not? The situation becomes even more enigmatic if we assume that Indian geometry derives from Greek geometry (see note 8, above), and that therefore some Indian mathematicians (presumably well before Aryabhata and Bhāskara) had been familiar with Euclidean procedures.
Here it is important to recall that Indian philosophers of the classical period were engaged in an ongoing debate with each other, in which radically different positions were defended and criticized. This debate went on because all participants were part of what I have called a
tradition of rational inquiry".which translated itself in the social obligation - partly embodied in kings and their courts - to listen to

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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

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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

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and mathematical literature. This may be brought out by compuring the grammar of Sanskrit with the geometry of Euclid - a particularly apposite comparison since. whereas mathematics grew out of philosophy in ancient Greece, it was, as we shall sec. partly an outcome of linguistic developments in India. The geometry of Euclid's Elements starts with a few definitions, axioms and postulates and then proceeds to build up an imposing structure of closely interlinked theorems, cach of which is in itself logically coherent and complete. In a similar fashion, Panini began his study of Sanskrit by taking about 1700 basic building blocks - some general concepts, vowels and consonants, nouns, pronouns and verbs, and so on- and proceeded to group them into various classes. With these roots and some appropriate suffixes and prefixes, he constructed compound words by a process not dissimilar to the way in which one specifies a function in modern mathematics. Consequently, the linguistic facility of the language came to be reflected in the character of mathematical literature and reasoning in India."
Maha-bh 1 p. 6 II. 3-7: katham tarkime sabdeh pratipattavydh kimeit suaminyavisesavar laksanam pravartyam yendipena yatena mahato mahatah sabdaughan pratipadyerukim puntas far utsargāpavadaw karcid utsargah karfayah kascid apavidal karhamjarvakah punar utsargah kartayyah kathamjatiyako pavidal samarrenotsarga kartavah. "Cp. Prakash Sarasvati, 1986: 157: "for the first time we find Aryabhata ... in his Aryabhatiya describing (mathematics) as a special section (Ganitaplida). Brahmagupta ... also followed Aryabhata in this respect and gave the science of calculation Ceanita) a special place in his treatise on astronomy. The Siddhanta treatises earlier than those of Aryabhata and Brahmagupta do not contain a chapter exclusively devoted to canita (the Sürya-Siddhanta and the Siddhantas of Vasistha, Pitamaha and Romaka are thus without ganita chapters)," Also Hayashi, 1995: 148: "Bhaskara (A.D. 629) tells us that Maskuri, Purana, Mudgala and other acaryas (teachers) composed mathematical treatises, but none of them is extant now. We can only have a glimpse of Indian mathematics of those early times through the extant astronomical works of Aryabhata, Bhaskara, and Brahmagupta, only a few chapters of which are devoted to mathematics proper. It should of course not be forgotten that much geometry is to be found in texts dealing with mathematical astronomy.
Aryabhatiya Bhasya p. 44 L 18: gamitam dviprakaram: rafiganitom ksetraganitom Cp. Hayashi, 1995: 62.
Aryabhatiya 2.17ab yas caiva bhajavargah kotivargas ca karnevargah sah. # Filliozat, 1988: 255-256. Another way of looking at the same characteristic is expressed in Pingree, 1978: 533, which speaks about the corrupt tradition of the earliest surviving Sanskrit texts: "The cause of this corruption is usually that the texts had become unintelligible; and this unintelligibility is not unrelated to the style developed by Indian astronomers. The texts proper were composed in verse in order to facilitate memorization, with various conventions for rendering numbers into metrical syllables. The exigencies of the meter often necessitated the omission of important parts of mathematical formulas, or contributed to the imprecision of the technical terminology by forcing the poet to substitute one term for another...."
Filliozat, 1995: 46 mentions the "high degree of generality" - which includes a general formulation of the Pythagorean theorem. "the first theorem enunciated in the history of Sanskrit mathematics" (p. 48) - as "a quality which a Sulbastra shares only with Panini model * Sharma, 1966, vol. III p. 829 Brāhmasphutasiddhanta 12.24) karnakse kotiktim risodra milam bhaje bhujasya krti prokya padam kotih kotibihurtytipadam karnahl. Tr. Colebrooke, 1817: 298. As a matter of fact, the Pythagorean theorem occurs several times, in several guises, in this portion of the text; e.g. 12.22cd: svavadhavargonad bhajavargdn mülam avalambak (tr. Colebrooke: "The square
root, extracted from the difference of the square of the side and square of its corresponding segment of the base, is the perpendicular"): 12.23cd: karakrtir bhimukhaytidalavargond padam lambah (tr. Colebrooke: "In any tetragon but a trapezium.) subtracting from the square of the diagonal the square of half the sum of the base and summit, the square root of the remainder is the perpendicular): 12.42ab: wdvysakriviesde milavyasantarardham isur alph (tr. Colebrooke: "Half the difference of the diameter and the root extracted from the difference of the squares of the diameter and the chord is the smaller arrow")
Bronkhorst, forthcoming 2: $ 2. "An exception may have to be made for the examples acompanying the sotras of the Bakhshall Manuscript
This is the name here adopted for the combined text consisting of the Daagliksetra vyakhya and the Aryabhata tantra-bhasya, expressions apparently used by Bhaskara himself to designate the two parts of his commentary (on chapter 1 and on the remaining chapters respectively see Shukla, 1976: xlix. 20 The Ganitaplida of Bhaskara's Aryabhatiya Bhasya has recently been studied and translated by Agathe Keller (2000). Reading her thesis has inspired me to write this article and helped me in the interpretation of many passages. a1 It is known that a number of classical commentaries imitated the style of the Mahabhisya, sometimes calling themselves "Vårttika" (Bronkhorst, 1990). Bhaskara's Bhasya does not adopt this so-called "Varttika-style" 22 Aryabhatiya Bhisya p. 96 L 15: vas cabhudargah war ca knivargah raw wargam ekatra karnavargo bhavari. 23 In order to avoid misunderstanding it must here be emphasized that I am aware of the fact that proof (Euclidean or other) does not appear to be essential to geometry in many cultures, see below. The present discussion on the absence of proof in Bhaskara and elsewhere finds its justification in the comparative approach adopted in this article, inspired by the remarks of Ingalls and Staal cited in its first pages. 24 Aryabhatiya Bhasya p. 44 L 17: kamabhujawah sematem karutisma atah karan, tr. Hayashi, 1995: 62. The dual is strange, and one wonders whether the original reading may not have been: bhujayoh karnasamavam karoti yamd fatah karanl. For a discussion of the term karanl in various mathematical texts, see Hayashi, 1995: 60-64 * Ganitapada 17cd: vitte sarasamvargo'rdhajyavargah sa thaly dhanusoh,
Aryabhatiya Bhasya p. 103 II. 12-13: pratyayakaranam ca sarvesy eva ksetresu "yas caiva bhudargah ko vargas ca karnawargah sah" iry anenaiveti. 27 Earlier authorities have already drawn attention to the absence of proofs in early classical Indian mathematics. So eg. Kline, 1972: 190, as cited in Srinivas, 1990: 76 n. 2: "There is much good procedure and technical facility, but no evidence that (the Hindus) considered proof at all." See however below. - Parallel to the hypothesis here considered to the extent that Panini may have exerted a negative influence on the development of certain sciences in India, is Lloyd's (1990: 87 ff.) observation that the influence of Euclid's Elements on the development of Greek science was not only positive. Lloyd draws attention in particular to medicine and physiology, certain areas of mathematics itsell, and to the extent to which problems in physics and elsewhere had a tendency to be idealized exactness may be obtained only at the cost of applicability)
This enumeration is not exhaustive even with regard to pusages identified in the text. Passages from the Mahabhasya are identified p. 3 II. 7-9, p. 8 II 1-2, p. 23 11. 25-26; but they are not mentioned in the appendix
It is tempting to see furthermore, in the line luke cu me i ganitapokaruh "wam wrddhimako 'pacaydimakova na bhavuti (p44 II 6-7, introducing Gunitapada 1) a reflection of Vkp 1.131ab: na so si pravayo lake yah kahdanwgamad nie.
62 The carattu jolie and Staaletines are
kative in pothesis fera
(1990: 32

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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?

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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

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able for
were not avere (Vogel. 1997
fora
that role was not alcation to the thesis we seem a merely and plays a role
ties in a decisive way influenced Egyptian, Greek, Indian and Chinese mathematics. in form as well as in catent, during the last half of the -Ist mill., if not earlier."
Lloyd sums up his views on the development of proof in the following passages (pp. 95-96): "We can say that the development, in Greece, of the demand for cen y sprung in part from a dissatisfaction shared by a variety of individuals with the merely persuasive.... We have related other intellectual developments that took place in early Greck thought to the political and social background, for example the extensive experience that many Greeks had of evaluating arguments in the law courts and assemblies.... We should conclude that in the development of formal or rigorous proof too... the political and legal background plays a role at least at the beginning of what might otherwise seem a merely intellectual development. However, the qualification to the thesis that must be entered is that, in this instance, that role was not as a source of positive, but rather of negative, models." " Cp Friberg. 1990: 583: 't is clear that the Greek mathematicians completely transformed the intellectual goods they received [from Babylonian mathematics).
Rigorous proofs based on abstract definitions and axioms took over the role played in Babylonian mathematics by a conceptually simpler method, that of using a reversal of the steps in an algorithm with given numerical data in order to check the computed values."
G.G. Joseph - in a chapter called "Egyptian and Babylonian mathematics: an assessment" - proposes to adjust the notion of proof so as to include these traditions (1991: 127): "A modern proof is a procedure, based on axiomatic deductido, which follows a chain of reasoning from the initial assumptions to the final conclusion. But is this not taking a highly restrictive view of what is proof? Could we not expand our definition to include, as suggested by Imre Lakatos ... explanations, justifications and elaborations of a conjecture constantly subjected to counter-examples? Is it not possible for an argument or proof to be expressed in thetoric rather than symbolic terms, and still be quite rigorous If one is determined to find proofs in all cultures that had geometry or something resembling it, adjusting the notion of proof may he the way to succeed; it is however open to doubt whether such a procedure adds much to our understanding
It appears that a copy of a Chinese translation of the Elements was present in the imperial library at the end of the thirteenth century, but was ignored; see Needham, 1959. 105 ff. Huff (1993: 241; with a reference to Aydin Sayili, The Observatory in Islam, Ankara 1960, p. 189) adds: "Even more tantalizing are the reports that a Mongol ruler in China, Mangu (d. 1257 ...) is said to have mastered difficult passages of Euclid by himself
Engelfriet, 1998: 297-298
Martzloff, 1997: 275. Yabuuti (2000: 40) suggests that Chinese mathematicians deduced theorems like that of Pythagoras by analogy, intuitively.
Martzloff, 1997: 72
Note further Marteloff. 1997: 276:"certain texts by Liu Hui (end of third century C.E.) and other mathematicians contain reasoning which, while it is not Euclidcan, is no less well constructed and completely exact. Moreover, although they are not numerous, these arguments appear all the more salient because they are without peers in other non-Euclidean mathematical traditions. But they also enable us to understand that Chinese mathematics is in part based on a small number of heuristic operational methods of a geometrical type... In fact, the most striking thing is the concrete appearance of the Chinese approach or rather the fact that abstract results af ved via concrete means. Chinese proofs tend to be based on visual or manual illustration of certain relationships rather than on purely discursive logic."
Bronkhorst. 1999 "C. Collins, 1908: 551: "Organizationally, the mathematiciuns, astronomers, and medical distors were based in private familistic lineages and guilds, never part
of the sustained argument provided by philosophical networks. Public networks of argument did exist in India, its philosophical lineages reached high levels of abstract development Only mathematics and science were not carried along with it." The striking absence of a Buddhist contribution to and participation in the development of astronomy and mathematics in classical India may be partly responsible for the relative peace" enjoyed by these branches of learning. (Vogel, 1997, shows that the Buddhists - or at least some of them were not averse to following developments in astronomy to fix the dates of their Posadha ceremony. The need to fix these dates did not apparently have the same effect as the Christian need to fix the date of Easter: about this later need Duncan, 1998: 79 states: "The history of science in the Middle Ages would have been very different if the bishops at Nicaea had decided to name a fixed date for Easter in the solar calendar"; further details in Heilbron, 1999.) Note that votisa/jyotis is mentioned in some lists of kalds occurring in Buddhist texts see Franco, 2000: 550 ([61]) with note 56. WA whole chapter of Brahmagupta's Brahmasphutasiddhanta (no. 11: Tantrapariksa) is dedicated to the refutation of different opinions.
For a recent example (from Jainism) of the close link between calendrical (ie. astronomical) and sectarian concerns, see Cort, 1999
He calls his teacher saratantranatantra and sarvesu antresu samom svatantra his father advaitavidyācārya (CESS 5 (1994), p. 735). " The one quotation from the Vakyapadiya (p. 22) concerns the meaning and function of upaya. Be it noted that Bhaskara's commentary on Gitikäpada 1 enumerates the five buddhindriyas, the five karmendriyas, plus manas, buddhi and sharikara, all of them known from Samkhya (p. 4 II. 11-15). Bhaskara also shows some acquaintance with Mimams: p. 182 I. 8 sarvasākhapratyayam kam karma may have been cited from Sabara's Bhasya on Mimamsasutra 2.4.9: the discussion immediately preceding this has a parallel in Sabara on 1.3.2
Mathematics is most often presented in treatises of astronomy, and it seems likely that astronomers often cared their living as astrologers. Cp. Al-Biron (E.C. Sachau's translation as reproduced in Chattopadhyaya, 1992: 510): "If a man wants to gain the title of an astronomer, he must not only know scientific or mathematical astronomy, but also astrology." Cp. Pingree, 1981: 56 as cited in Yano, 1987: 54: "There was never in India a jati(caste, MY] of mathematicians, and rarely anything that could be called a school: most mathematicians were joriy (astronomers or astrologers)." Pingree (1993: 77) argues that Aryabhata, far from making observations himself, derived the longitudes of the planets "from astrological playing with numbers". "On the low esteem in which astrologers were held, see Kane, 1974: 526 ff.
Aryabhatiya Bhāsya p. 481. 16 ff. The edition of K.S. Shukla contains a somewhat different diagram; however, a manuscript page reproduced in Keller's thesis (1 p. 223) appears to support her construction " Cp. Srinivas, 1990: 35; Sarasvati Amma, 1999. 133 ff. Keller refers in this connection to an unpublished Ph.D. thesis of Simon Fraser University: A critical edition, English translation and commentary of the Upodghāta Şad vidhaprakarana and kutakadhikara of the Suryaprakasa of Süryadasa, by Pushpa Kumari Jain, 1995. " David Pingree (private communication) points out that certain theorems on cyclic quadrilaterals presented by Brahmagupla (628 C.E.) were not developed from a Euclidean approach until the 16th and 17th centuries in Europe.

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