Tattvopaplavavāda of Jayarāśi and its Alleged Relation to the Cārvāka/Lokāyata

Ramkrishna Bhattacharya

The Lion of Upsetting of all Principles (Tattvopaplava-siṃha) by Jayarāśibhaṭṭa has been claimed by some scholars to be the only surviving Cārvāka work. Others have challenged this view.ⁱ Since there is no external evidence to settle the question, the debate continues solely on the basis of internal evidence and intrinsic probability. No near-unanimous (total unanimity is seldom found) conclusion has been reached to date. Instead of summarizing the whole debate (which is of a technical nature), a few issues are raised below. They go against branding Jayarāśi a Cārvāka, but identify him rather as a founder/follower of a totally new doctrine, which is quite distinct from both materialism on the one hand and any form of illusionism (māyāvāda) or nihilism (śūnyavāda) on the other. His is ‘the doctrine of upsetting principles,’ tattvopaplava-vāda. This is the name used by Jayarāśi’s critics; nowhere is he called a Cārvāka or ‘one belonging to a section of the Cārvāka’ (Cārvākaikadeśīya).ⁱⁱ It is basically an annihilationist doctrine. Although Jayarāśi is called a skeptic, there is no room for such a thing as doubt in his work. He is convinced that there can be no principles, tattvas, because there is no such thing as pramāṇa.  

Now, Vātsyāyana in the exordium of his Nyāya Aphorisms (Nyāyasūtra) states at the outset (on 1.1.1) that one has to admit not only pramāṇa, but also the other three, pramātṛ (knower), prameya (the object rightly known), and pramiti (right knowledge of the object): ‘With these four, tattva reaches its fulfillment.’ M.K. Gangopadhyaya suggests that, as opposed to tattvopaplavavāda, Vātsyāyana’s view may be called ‘the doctrine of establishing the principles,’ tattva-vyavasthāpana-vāda.ⁱⁱⁱ It seems Vātsyāyana had a predecessor of Jayarāśi in mind, and against such an opponent he felt it necessary to assert all the four elements stated above. This assertion can be understood only against the backdrop of an opponent who denied pramāṇa as such. 

‘Skepticism’ has a definite significance in western philosophy; it is improper to use it in the Indian context.^iv A skeptic has no axe to grind; he or she merely doubts the veracity of all views. Jayarāśi however has a definite view of his own. At the end of his work he claims that even those (questions) which could not become the object of knowledge of even the preceptor of the gods have been raised by Bhatta Sri Jayarāśi, for the shake of removing the pride of the infidels.^v On the basis of this declaration, and the Cārvāka aphorisms quoted at the beginning of the work, he has been called a Bārhaspatya (follower of Bṛhaspati, the legendary founder of materialism) or a Cārvāka/ Lokāyata. To this facile identification D. Chattopadhyaya objects, ‘[A]ccording to the Indian philosophical tradition no real representative of a system would ever dream of boasting intellectual superiority to the founder of the system itself. Jayarāśi, who claims to be intellectually superior to Bṛhaspati, could thus hardly be a follower of Bṛhaspati himself, i.e., could hardly be the leader of any imaginary offshoot of the Cārvāka or Bārhaspatya system.’^vi 

Gangopadhyaya endorses this view and adds: the way Jayarāśi uses honorific plural in mentioning his own name along with Bṛhaspati, bhaṭṭaśrījayarāśi-devagurubhiḥ…, places him in the seat of the preceptor of the gods, which goes against the Indian tradition. Jayarāśi further claims that all his opponents will be defeated by his arguments. This too is not the style of the explicators of Indian philosophy. The way of writing of later writers, even if they express views of their own, is suave and modest, as if they mean to suggest that this significance was inherent in the text itself.^vii 

Chattopadhyaya adduced another argument against the identification of Jayarāśi as a Cārvāka:

It is moreover necessary to remember that Jayarāśi claims as his final achievement the annihilation of the vanity of the Pāṣaṇḍin [pākhaṇḍin]-s ([Tattvopaplava-siṃha Baroda ed.] p.125). Now whatever might have been the exact meaning of the word pāṣaṇḍin,^viii it could by no stretch of imagination have excluded the Lokāyatikas and Cārvākas.^ix

Gangopadhyaya further elucidates:

According to the lexicographers pā in the word pākhaṇḍa means trayīdharma [i.e., the Vedas]; those who refute (khaṇḍayati) that dharma are pākhaṇḍa, i.e. anti-Vedic nāstikas. The Cārvākas in Indian philosophy are well-known by the appellation nāstikaśiromaṇi [vide Sarva-darśana-saṃgraha, ed. Abhyankar, chap.1, p.2]. Then how can Jayarāśi be a supporter of Cārvāka?^x

The derivation of pākhaṇḍa is of course an instance of folk etymology, but such derivations, however absurd, are pointers to actual usage. Unless pākhaṇḍa was once current in this sense, such a derivation would not be proposed at all. The etymology is required to conform to common understanding, justifying the sense in which it was already in use.

Did the Cārvākas or the earlier bhūtavādins and lokāyatas (mentioned in the Tamil epic, Manimekalai, 27.264-277) call themselves bārhaspatyas, thereby giving a stamp of approval to the Purāṇic origin of their philosophical system which in its turn takes its cue from a legend in the Maitrī Upaniṣad (7.9)? No available fragment suggests so. On the other hand, from Purandara we learn that the new materialists used to call themselves Cārvākas and we also know from other sources that the base text of their school was known as the Paurandaraṃ sūtram, and there was also a commentary (an auto-commentary) called Pauraṃdariya vitti as well.^xi The very idea of referring to their origin to the suraguru (devaguru) or calling him bhagavān, as found in theTattvopaplava-siṃha, would be anathema to them.

There is indeed a Cārvāka at the very beginning of the Tattvopaplava-siṃha. But he is not Jayarāśi, but another person who is presented as a Cārvāka out to challenge Jayarāśi’s doctrine of upsetting tattva as such. This objector has to be a Cārvāka, for who but a Cārvāka would refer to the basic premises of materialism and stand upon them? The presence of this objector and the way Jayarāśi gets into controversy with him clearly indicate that Jayarāśi himself was not a Cārvāka or did not even belong to ‘a section of the Cārvāka’ (cārvākaikadeśya). He prided in claiming that he could understand Bṛhaspati’s sūtras better than the Cārvākas themselves. Jayarāśi always refers to Bṛhaspati, the mythical guru of the gods, never to a real-life philosopher like Purandara or Aviddhakarṇa, as Kamalaśīla, Karṇakagomin, Anantavīrya, Cakradhara, and Vādidevasūri do. Thus Jayarāśi supports the purāṇic story of the origin of materialism. Sāyaṇa-Mādhava too refers to Bṛhaspati as the author of a number of verses that are found in the Viṣṇu Purāṇa and Buddhist and Jain sources.^xii

 ^Notes:

i Eli Franco (Perception, Knowledge and Disbelief: A Study of Jayarāśi’s Scepticism (Delhi: MLBD, 1994) )modifies this assertion by calling Jayarāśi a sceptic Lokāyata rather than a materialist (XII-XIII), but very few pay attention to his distinction. They call Jayarāśi a Cārvāka or a Lokāyata, apparently meaning a materialist.

ii For instance, Vidyānandasvāmin, Aṣṭasahasrī (Mumbapuri: Nirnayasagara Press, 1915) 37: tadime tattvopaplavavādinaḥ… ; idem,Tattvārthaślokavārttika (Mumbapuri: Nirnayasagara Press ,1918) 80, 195; Anantavīrya, Siddhiviniścayaṭīkā (Kashi: Bharatiya Jnanapith,1959) 277-278 – all treat the Cārvāka and tattvopaplava-vāda separately. – For a survey of the Jayarāśi controversy, see Sukhlalji Sanghvi and Rasiklal Parikh, Introduction to the Tattvopaplavasiṃha of Jayarāśibhaṭṭa (Baroda: Oriental Institute, 1940) i-xiv, reprinted in Cārvāka/Lokāyata (n28 above), 492-504 and Eli Franco (n34 above), XI-XIII, 4-8. For the other view see Walter Ruben (reprinted in Cārvāka/Lokāyata, 505-519), and K. K. Dixit (reprinted in Cārvāka/Lokāyata, 520-530), D. Chattopadhyaya, Indian Philosophy: A Popular Introduction (New Delhi: People’s Publishing House, 1964), 222-223, and In Defence of Materialism in Ancient India (New Delhi: People’s Publishing House, 1989) 36-41. See also R. Bhattacharya in an interview with Krishna Del Toso, “The Wolf’s Footprints: Indian Materialism in perspective,” Annali 71 (2011): 183-204 (particularly 188-191), and Tattvopaplavasiṃha of Jayarāśibhaṭṭa, trans. V. N. Jha (Ernakulam: Chinmaya International Foundation Shodha Sansthan, 2013) xi. 

 

iii “Mukhavandha” (Foreword ) to D. K. Mohanta , Tattvopaplavasiṃha: Jayarāśibhaṭṭer Saṃśayavāda ( Kolkata: Sanskrita Sahitya Bhandar, 1998) [xiii]. 

 

iv It is also improper to translate nāstika as ‘agnostic’ (as Goldman has done in his translation (The Rāmāyaṇa of Vālmīki, vol. 1, Delhi: Oxford University Press, 1984) 136), for a nāstika in its earliest use is a non-believer in the other-world, later one who denies the authority of the Veda; then one who says there is no god or gods (atheist). In short he or she is a typical neinsager (one who says no). But he or she is neither a doubter nor undecided about acquisition of knowledge as such. See R. Bhattacharya, “Development of Materialism in India: the Pre-Cārvākas and the Cārvākas,” Esercizi Filosofici 8 (2013): 1-12.

v Sanghvi and Parikh (see n34 above) 124; V. N. Jha (n34 above) 464.

vi Indian Philosophy: A Popular Introduction, (see n34 above) 222-23. Even earlier, in two essays in Bangla published in 1963 (see his Saṃghaṃ Śaraṇaṃ Gacchami ityādi agranthita racanā (Kolkata: Ababhas, 2010) 74-84) Chattopadhyaya stated the same point. 

 

vii M. K. Gangopadhyaya , “Mukhavandha,” [xi].

viii The Śabda-kalpa-druma (Kalikata: Hitavadi Karyyalaya, 1836 Saka era (1924 CE)) glosses pāṣaṇḍa as one who behaves contrary to the Veda, one who bears all the marks (i.e., a heretic) ; the Buddhists, the Jains and others (some commentators on the lexicons name these two communities in particular); The Sanskrit-Wörterbuch (Eds. O. Böhtlingk and R. Roth (Delhi: MLBD, 1990 reprint)) glosses pāṣaṇḍa as Irrlehre (untaught), Ketzerei (heresy), also Ketzer (heretic).

ix D. Chattopadhyaya, Indian Philosophy, 223.

x M. K. Gangopadhyaya, Foreword, [xi]. The etymology of pākhaṇḍa as suggested is found in an anonymous verse quoted in Bhānuji Dīkṣita’s Rāmāśramī (Vyākhyāsudhā), a commentary on the Nāmaliṃgānuśasana (Amarakoṣa) (Varanasi: Chaukhambha Sanskrit Sansthan, Vikrama era 2099), Brahmavarga, 44d.

xi R. Bhattacharya, Studies, Chap. 5.

Bibliography

Ācāraṅgasūtram and Sūtrakṛtāṅgasūtram with Niryukti of Ācārya Bhadravāhu Svāmī and the Commentary of Śīlāṅkācārya. Ed. Ācārya Sarvanandājī Mahārāja. Re-ed. with Appendix by Muni Jambuvijayaji. Delhi: MLBD Indological Trust, 1978.

Anantavīrya, Siddhiviniścayaṭīkā . Kashi: Bharatiya Jnanapith,1959.

Bhānuji Dīkṣita. Rāmāśramī (Vyākhyāsudhā), a commentary on the Nāmaliṃgānuśāsana (Amarakoṣa). Varanasi: Chaukhambha Sanskrit Sansthan, Vikrama era 2099.

Bhattacharya, Ramkrishna, interview with Krishna Del Toso. “The Wolf’s Footprints: Indian Materialism in perspective.” Annali 71 (2011): 183-204.

Bhattacharya, Ramkrishna. “Pre-Cārvāka Materialism in Vasudevahiṃdī.” Jain Journal 43(3) (2009): 102-109.

Bhattacharya, Ramkrishna. “Svabhāvavāda and the Cārvāka/Lokāyata: A Historical Overview.” Journal of Indian Philosophy 40(5) (2012): 593-614.

Bhattacharya, Ramkrishna. “The Case of Āryabhaṭa and His Detractors.” The Indian Historical Review 17 (1990-91): 35-47.

Bhattacharya, Ramkrishna. “The First Cause: Rivals of God in Ancient Indian Thought.” Indian Skeptic 14 (2001a): 19-23.

Bhattacharya, Ramkrishna. “The first Cause: Syncretic Bias of Haribhadra and Others.” Jain Journal 35 (3-4) (2001b): 179-184.

Bhattacharya, Ramkrishna. “The Social Outlook of the Cārvāka/Lokāyata: A Reconstruction.” Indologica Taurinensia 36 (2010): 37-42.

Bhattacharya, Ramkrishna. “Verses Attributed to Bṛhaspati in the Sarvadarśanasaṃgraha: A Critical Appraisal.” Journal of Indian Philosophy 41 (2013): 615-630.

Bhattacharya, Ramkrishna. Studies on the Cārvāka/Lokāyata. Firenze: Società Editriche Fiorentina, 2009; London: Anthem Press, 2011.

Böhtlingk, Otto und Rudolf Roth. Sanskrit-Wörterbuch. Delhi: Motilal Banarsidass, 1990 reprint.

Brahmajālasutta. Dīghanikāya. Patna: Pali Publication Board, 1958.

Cārvāka/Lokāyata. Eds. D. Chattopadhyaya and M. K. Gangopadhyaya. New Delhi: Indian Council of Philosophical Research, 1990.

Chattopadhyaya, Debiprasad. Indian Philosophy: A Popular Introduction. New Delhi: People’s Publishing House, 1965.

Chattopadhyaya, Debiprasad. In Defence of Materialism in Ancient India. New Delhi: People’s Publishing House, 1989.

Chattopadhyaya, Debiprasad. Saṃghaṃ Śaraṇaṃ Gacchami ityādi agranthita racanā. Kolkata: Ababhas, 2010.

Cimanabhaṭṭa. Āryavidyāsudhākara. Lahore: Moti Lal Banarsi Das, 1923.

Dahlmann, Joseph. Die Sāṃkhya-Philosophie. (Berlin: Felix L. Demes, 1902) qtd. inThe Mahābhārata. Critical Edition. Śāntiparvan. Ed. S. K. Belvalkar. Poona: Bhandarkar Oriental Research Institute, 1951.

Del Toso, K., "tebhyaś caitanyaṃ : il 'sé' secondo il Materialismo indiano." In Intrecci filosofici. Pensare il Sé a Oriente e a Occidente, edited by A. Cislaghi and K. Del Toso, pp. 135-153. Milano-Udine: Mimesis, 2012,.

Deva, Radhakanta. Śabdakalpa A. Cislaghi and K. Del Toso (ed.), druma. Kalikata: Hitavadi Karyyalaya, 1836 Saka era (1924 CE).

Dharmakīrti, auto-commentary on the Pramāṇa-Vārttika. Ilahabad: Kitab Mahal, 1943.

Dixit, K. K. “The Ideological Affiliation of Jayarāśi – The Author of Tattvopaplavasiṃha.” In Cārvāka/Lokāyata, edited by D. Chattopadhyaya and M. K. Gangopadhyaya, pp. 520-530. New Delhi: Indian Council of Philosophical Research, 1990.

Franco, Eli. Perception, Knowledge, Disbeilief / A Study on Jayarāśi’s Scepticism. Delhi: Motilal Banarsidass, 1994.

Frawallner, Erich. History of Indian Philosophy. Vols. 1 and 2. Delhi: Motilal Banarsidass, 1971.

Gangopadhyaya, M. K. Mukhabandha (Foreword) to Dilip Kumar Mohanta. Tattvopaplavasiṃha: Jayarāśibhaṭṭer Saṃśayavāda. Kalikata: Sanskrita Pustak Bhandar, 1998.

Gerschhiemer, Gerdi. “Les ‘Six doctrines de spéculation’ (ṣaṭtarkī) – Sur la categorization variable des systems philosophiques dans l’Inde classique.” In Expanding and Merging Horizons.Wilhelm Halbfass Memorial Volume, edited by Karin Preisendanz, pp.239-258. Vienna: Austrian Academy of Sciences Press, 2007.

Haribhadra. Ṣaḍdarśanasamuccaya with Guṇaratna’s and Somatilakasῡri’s commentaries and an anonymous Avacῡrṇi. Calcutta: Bharatiya Jnanapitha, 1969.

Harvey, Paul. Oxford Companion to Classical Literature. Oxford: Oxford University Press, 1984.

Horace. Epistles. London: William Heinemann, 1926.

Jamkhedkar, A. P. Vasudevahiṃḍī: A Cultural Study. Delhi: Agam Kala Prakashan, 1984.

Jayantabhaṭṭa. Nyāyamañjarī. Part 1. Varanasi: Sampurnanand Sanskrit Visvavidyalaya, 1982.

Jayarāśibhaṭṭa.Tattvopaplavasiṃha of Jayarāśibhaṭṭa. Baroda: Oriental Institute, 1940.

Jayarāśibhaṭṭa. Tattvopaplavasiṃha of Jayarāśibhaṭṭa. Trans. V. N. Jha. Ernakulam: Chinmaya International Foundation Shodha Sansthans, 2013.

Jinendrabuddhi.Viśālāmalavatī Pramāṇasamuccayaṭīkā. Eds. Ernst Steinkellner et al. Beijing: China Tibetology Publishing House and Vienna: Austrian Academy of Sciences Press, 2005.

Jones, W. H. S. Hippocrates. Vol.1. London: William Heinemann, 1972.

Madhusudana Sarasvatī. Prasthānabhedaḥ. Poona: Anandashrama, 1877.

Manu-smṛti with Nine Commentaries. Vol. 2. Ed. J. H. Dave. Mumbai: Bharatiya Vidya Bhavan, 1975.

Müller, F. Max. Lectures on the Origin and Growth of Religion. London: Longmans, Green & Company, 1891 reprint.

Müller, F. Max. The Six Systems of Indian Philosophy (1899). Varanasi: The Chowkhamba Sanskrit Series Office, 1971.

Patañjali’s Mahābhāṣya. Paśpaśāhnika. (1972). Ed. with English translation, etc. Kshitish Chandra Chatterji. Calcutta: A. Mukherjee & Co.

Patañjali’s Vyākaraṇa-Mahābhāṣya, Āhnikas 1-3. (1975). Ed. K. V. Abhyankar and Jaydev Mohanlal Shukla. Poona: Bhandarkar Oriental Research Institute.

Radhakrishnan, S. and C. A. Moore A Source Book in Indian Philosophy. Princeton: Princeton University Press, [1957].

Rāmāyaṇa of Vālmīki,The. Vol. 1, trans. Robert P. Goldman. Delhi: Oxford University Press, 1984.

Rhys Davids, T. W. Introduction to the Mahānidānasuttanta. Dialogues of the Buddha. London: Oxford University Press, 1910.

Ruben, Walter. “On the Tattvopaplavasiṃha of Jayarāśibhaṭṭa.” In Cārvāka/Lokāyata, edited by D. Chattopadhyaya and M. K. Gangopadhyaya, pp. 505-519. New Delhi: Indian Council of Philosophical Research, 1990.

Sanghvi, Sukhlalji and Rasiklal Parikh. Introduction to the Tattvopaplavasiṃha of Jayarāśibhaṭṭa. Baroda: Oriental Institute, 1940. Reprinted in Cārvāka/Lokāyata, edited by D. Chattopadhyaya and M.K.Gangopadhyaya, pp. 492-504. New Delhi: Indian Council of Philosophical Research, 1990.

Saṅghadāsagaṇi Vācaka. Vasudevahiṃḍī Prathama Khaṇḍam, eds. Caturavijaya and Punyavijaya (1930-31)

Gandhinagar: Gujarat Sahitya Akademi, 1989.

Śaṅkarācārya. Brahmasūtrabhāṣya. Delhi: Motilal Banarsidass, 1988.

Śāntarakṣita. Tattvasaṃgraha with Pañjikā by Kamalaśīla. Varanasi: Bauddha Bharati, 1968, 1981.

Sarup, Lakshman (ed.). The Nighaṇṭu and the Nirukta. Delhi: Motilal Banarsidass, 1984.

Sastri, T. Ganapati. Arthaśāstra of Kauṭalya with the Śrīmūla Commentary (1924). Dilli: Bharatiya Vidya Prakashan, 1984.

Selsam, Howard and Harry Martel. Reader in Marxist Philosophy. New York: International Publishers, 1987.

Shastri, D. R. A Short History of Indian Materialism, Sensationalism and Hedonism (1930). Calcutta:

Bookland, 1957. Reprinted in a slightly abridged form in Cārvāka/Lokāyata, edited by D. Chattopadhyaya

and M. K. Gangop

Ramkrishna Bhattacharya taught English at the University of Calcutta, Kolkata and was an Emeritus Fellow of University Grants Commission. He is now a Fellow of PAVLOV Institute, Kolkata.

Email ID: ramkrishna.bhattacharya@gmail.com)

adhyaya, pp. 394-43. New Delhi: Indian Council of Philosophical Research, 1990.

Vidyānandasvāmin. Aṣṭasahasrī. Mumbapuri: Nirnaysagara Press, 1915.

Vidyānandasvāmin. Tattvārthaślokavārttika. Mumbapuri: Nirnayasagara Press ,1918.

Vyomaśiva. Vyomavatī. Varanasi: Sampurnanand Sanskrit Vishvavidyalaya, 1983-84.

Read More

The Home of the So-called Pythagorean Theorem: Babylonia, Mesopotamia, China, India,… Vardhan’s and Tharoor’s Claims Refuted

Ramkrishna Bhattacharya

In a recent speech delivered before the scientists attending the 102nd edition of the Indian Science Congress, Dr. Harsh Vardhan is reported to have said: "Our scientists discovered the Pythagoras theorem but we very sophisticatedly [!] gave its credit to the Greeks' (The Times of India, 08 January 2015). Mr. Vardhan is not a non-entity; he is a Hindutvavadi Bharatiya Janata Party (BJP) stalwart and at present the Union Minister of Science and Technology. Mr. Shashi Tharoor, a Congress Member of Parliament (Lok Sabha), shelving his opposition to the ruling BJP, supported Vardhan in a series of tweets. He said, '... the Sulba Sutras, composed between 800 and 500 BC, demonstrate that India had Pythagorean theorem before the great Greek was born' (The Hindu, 08 January 2015).

  

Bust of Pythagoras of Samos in
the Capitoline Museums, Rome.

There is nothing new in this claim. Dr. George Thibaut, the scholar who first studied the Sulbasutras in detail and translated one of them, had mentioned the Pythagorean Theorem in relation to the Sulba geometry as early as 1875 in an long essay published in the Journal of the Asiatic Society of Bengal. All histories of mathematics in general and of geometry in particular also recognize this similarity. But what the hon’ble minister and the opposition MP have stated is a curious mixture of half-truths and downright lies. Let us see where both of them went wrong.

A few words about the Sulva (or Sulba)-sutra first. It is admitted on all hands that the rudiments of geometry in India developed from that branch of the six Vedanga-s (lit. ‘limbs of the Veda’) called Kalpa. Vedic priests connected with the Yajurveda were supposed to learn this particular ancillary literature of the Veda. The Sulbasutras form a part of this Kalpa. They deal with, among other things, the piling of the fire altar, variously called Agni, Cayana, Citi, and Vedi, required for the Soma sacrifice (yaaga). The altars were made of kiln-burnt bricks. The bricks are of different shapes and sizes and required the skill and experience of the manual workers (not sages or professional scientists), such as, brick-makers and masons.

And here is the crux. The masons used ropes and bamboo in their work. The word sulba means ‘rope’. In the texts of the Sulbasutras of different schools, the word rajju (cord) is used throughout. They had no ruler and compass, and so all operations had to be made and measured with ropes and bamboos. There was no geometer, no scientist when the bricks were burnt and Vedis were piled. Whatever knowledge was acquired from such operations were essentially empirical in nature. Therefore, the very idea of a theorem is not to be expected. So Mr. Vardhan’s praise of ‘our scientists’ is beside the point. 

 

Now let us look at the Sulba texts in which the so-called Pythagorean Theorem is suggested:

(a) ‘The cord stretched across a square (i.e. in the diagonal) produces an area of the double size’. (Baudhayana Sulbasutra, 1.45)

(b) ‘The diagonal of an oblong produces by itself both the areas which the two sides of the oblong produce separately’ (i.e., the square of the diagonal is equal to the sum of the squares of the two sides). (Baudhayana Sulbasutra, 1.48)

Why state the same in two different aphorisms(sutra)s? We must remember one significant fact: the Sulbasutras recognize only the caturasra, quadrilateral. There is no concept of the trilateral as such. The Pythagorean Theorem, however, is concerned with the right-angled triangle only. It runs as follows: ‘In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.’ There was, however, no concept of angles and their measurement by degrees in ancient India. Only one type of trilateral is referred to by name: Praüga, the isosceles trilateral. Praüga is the name of the fore part of the shafts of a chariot. An altar is also named after it. The word, tisra occurs in relation to a particular kind of brick (Baudhayana Sulbasutra, 4.61), a right-angledd trilateral. The Manava Sulbasutra mentions trikona (10.3.7.6) but one side of this trigonal brick is curved. In any case the word, kona, does not stand for ‘angle’ (as it normally does today in many North Indian languages), but simply means ‘corner’. Thus panchakona in Manava Sulbasutra, 10.3.7.6 suggests a five-cornered figure. 

 

But there is no common name to suggest the trilateral as such. The right-angled trilateral is always conceived as a semi-quadrilateral – specifically a square (sama-caturasra) or an oblong (dirgha -caturasra) halved by a diagonal. There is no concept of the angle and hence, of the triangle in the Sulbasutras. The word, ‘triangle’ is quite inappropriate in the world of Vedic sacrifices. They knew only the quadrilateral, called chaturasra. Squares, rectangles and other quadrilateral figures were there, for the bricks were made of such shapes first, and then divided into several parts. Thus we have the figures, that are but the shapes of the bricks employed in piling the altar, all derived from a square as shown below: 

 

In the same way, the oblong (rectangle) too was divided into several such parts and each was accorded the name of its own. 

 

Hence the so-called Pythagorean Theorem is stated twice in the Baudhayana Sulbasutra: first in terms of a square [Proposition (a)] and then in terms of an oblong [Proposition (b)].

One may object that even though the trilateral was formed out of a square or an oblong, it was there, and therefore, there is nothing to prevent us from claiming that the so-called Pythagorean Theorem was ‘discovered’ by the Yajurvedic priests. 

 

Unfortunately, the texts of the Sulbasutras in which the statement resembling the Pythagorean Theorem occurs gives a lie to such a claim.

Some examples of the application of the proposition are also provided in another sutra: in connection with an oblong the sides of which are 3 and 4, 15 and 8, 7 and 24, 12 and 35, 15 and 36 (Baudhayana Sulbasutra, 1.49). One can easily see the relationship between the sum of the squares on the base and the perpendicular being equal to the square on the hypotenuse.

Thus, 3² + 4²= 5², 15² + 8²= 17², 7² + 24²= 25², 12² + 35²= 37², 15² + 36²= 39².

The same proposition also occurs in Apastamba Sulbasutra, 5.5, etc. and Katyayana Sulbasutra, 2.11. But no attempt at generalization is ever made. 

 

Any attempt to prove that the so-called Pythagorean Theorem (Proposition 1.47 in Euclid’s Stoikhna, in English Elements) was known in that very form in India before Pythagoras (flourished about 530 bce) is futile. First of all, the dating of the Sulbasutras is conjectural, but it cannot be earlier than the 600 bce. The dating of ancient Indian texts is always problematic: unanimity of scholarly opinion is seldom to be expected. (We have followed the chronological table given in the opening pages of A. N. Ghatage and others (eds.) An Encyclopedic Dictionary of Sanskrit on Historical Principles. Poona: Deccan College, Volume I, 1978).

In any case, such claims and counterclaims have long ago become meaningless, since it has been decisively proved that the ‘theorem’ was known in Old Babylonia at least twelve hundred years before Pythagoras. (A. Seidenberg, ‘The Geometry of the Vedic Rituals’ in: Agni, Vol. 2, edited by Frits Staal with assistance of Pamela MacFerland, Berkeley: Asian Humanities Press, 1983, p. 101. For the Cuneiform texts containing Pythagorean numbers (and triples), see Midonick (ed.), Treasury of Mathematics. Harmondsworth: Penguin Books, Vol. 1, 1968, pp. 29-35). Both Mesopotamia and China too have a claim in having a glimpse of the Pythagorean triples. (Carl Benjamin Boyer (1968). "China and India". A history of mathematics. Wiley. p. 229; S. N. Sen and A. K. Bag, The Sulbasutras. New Delhi: Indian National Science Academy, 1983, pp. 10-11, 154). Even if the redactors of the Satapatha Brahmaṇa were aware of the theorem (as Seidenberg says, p. 106), the work cannot be pushed back to 1900-1600 bce. We should rather note that the theorem was formulated in India out of the practice of craftsmen quite independently of Babylonia or Greece or China. The same statement would be true of Greece and China as well. Here is an excellent example of polygenesis: the same conclusion was arrived at in different ancient civilizations, unbeknown to one another, all on the basis of empirical observation. Euclid provided a general theorem true for all right-angled triangles and attributed it to Pythagoras, and there lies his credit. His predecessors could only think of and note down several cases, but stopped there.

 ***

 

Dr Ramkrishna Bhattacharya taught English at the University of Calcutta, Kolkata and was an Emeritus Fellow of University Grants Commission. He is now a Fellow of Pavlov Institute, Kolkata.

Read More