Studies of India’s ancient scientific accomplishments have seen two extremes: At one end of the spectrum, daydreamers fancy that the Vedas knew everything from electricity to interplanetary travel, that vimānas crisscrossed Indian skies millenniums ago, or that Aryabhata invented all mathematics. At the other end, gainsayers bristle at the thought that some science might not have emerged from the “Greek miracle”: Indian scientific advances can only be borrowed or derivative, its imperfections and errors alone being original contributions, while its rational elements ultimately stem from contact with the Greeks; Indian savants knew no experimental science, followed no proper axiomatic method, and in any case ended up in stagnation, while Europe galloped forth triumphantly and gave us the boon of “modern science”.
With minor variations and boring predictability, the two scenarios are repeated decade after decade, while serious scholars — both Indian and Western — quietly and patiently generate solid material which, in a normal (rational?) world, should suffice to dismiss dreamers and gainsayers alike to the obscurity they deserve. Indeed, ridiculing the former is easy, and occasionally needs to be done. Exposing the latter, however, is less commonly done, as they often conceal their biases or ignorance behind academic posts and imposing jargon.
A recent case in point is Meera Nanda, who has been for some years on a self-appointed mission to expose all claims to knowledge by (let us lump them together, as she does) Hindu enthusiasts, nationalists, right-wingers or Hindutva activists. Her latest contribution, “Hindutva’s science envy” (Frontline, August 31), blames in a vast sweep “the current crop of Hindu nationalists and their intellectual enablers” for being the progeny of thinkers like “Bankimchandra Chattopadhyaya, Vivekananda, Dayananda Saraswati, Annie Besant (and fellow Theosophists), Sarvepalli Radhakrishanan, M.S. Golwalkar and countless other gurus, philosophers and propagandists” — doubtless a most despicable crowd!
I will not deal with Nanda’s personal attacks on past and present figures, but will confine myself to discussing the considerable distortions in her two case studies of Indian mathematics: the case for early Indian knowledge of the Pythagoras theorem, and India’s claim to be “the birthplace of the sunya, or zero.”
The Pythagoras Theorem
Nanda attacks the view that Baudhāyana’s Shulbasūtras, a text of geometry for the construction of fire altars, which she dates “anywhere between 800 and 200 bce,” knew the Pythagoras theorem (on right-angled triangles) before the Greek savant himself. Thus, Nanda informs us, Pythagoras “comes in for a lot of abuse in India.” But this view is not that of hot-headed enthusiasts; it was stated as early as in 1822 by the British astronomer John Playfair (“On the Astronomy of the Brahmins”): “It is curious to find the theorem of Pythagoras in India, where, for aught we know, it may have been discovered.” The eminent historian of Indian mathematics Bibhutibhushan Datta (in his Ancient Hindu Geometry of 1932) showed that knowledge of the “theorem” was actually traceable to the much earlier Taittirīya Samhitā (also known as Krishna Yajurveda) and Shatapatha Brāhmana, the first of which dates back to 1000 bce at the least. In his landmark 1960 paper on “The Ritual Origin of Geometry”, the U.S. mathematician and historian of mathematics A. Seidenberg independently reached similar conclusions: “The Pythagoras theorem ... was known and applied at the time of the Taittiriya Samhita.” Playfair, Datta or Seidenberg were not members of the Sangh Parivar, to my knowledge; neither can they be blamed for “abusing” Pythagoras.
Nanda proceeds to ridicule the thesis that the Greek savant might have come to India to learn geometry “from Hindu gurus,” unaware that the said thesis emerged not from one of her bêtes noires, but from a few minor neo-Platonic Greek texts picked up and amplified by Enlightenment philosophers such as Voltaire, the French astronomer Jean-Sylvain Bailly (in 1777) or the British Edward Strachey (1813).
Nanda then points out that the Mesopotamians knew the theorem about 1800 bce, which “blows holes through much of the case for Baudhayana’s priority.” Strictly speaking, all that the Mesopotamian tablets in question show is an acquaintance with certain sets of Pythagorean triplets; this may or may not imply knowledge of the theorem in its general form (as given in the Shulbasūtras). Even if conceding that the Mesopotamians did know that general form, as is likely, does this badly puncture the Indian text’s “priority”? Not necessarily, since the Shulbasūtras enshrine a geometrical tradition much older than the texts themselves, as Datta and Seidenberg demonstrated. How much older is a matter of speculation in the absence of clinching evidence.
But why should “priority” matter so much, after all? Nanda does not cite a single serious scholar, not even a “nationalist” one, who worries about it. Historians of mathematics rightly prefer to concentrate on understanding how each geometrical tradition — Mesopotamian, Greek, Indian or Chinese — approached and applied the theorem, or whether (as Seidenberg concluded) the first three traditions had a common origin.
Pythagoras in China
Nanda then informs us — and this is supposed to be very damaging — that the first “proof” of the Pythagoras theorem is found not in the Shulbasūtras but in a Chinese text of unknown authorship, Chou Pei Suan Ching, “dated anywhere from 1100 to 600 bce.” In current spelling, this is the Zhou Bi Suan Jing (“Mathematical Classic of the Zhou Gnomon”), which was “most probably compiled no later than the first century bce,” according to Joseph W. Dauben, a distinguished historian of science and expert on Chinese mathematics (I borrow his translation of the work’s title). Christopher Cullen, another respected expert, agrees that the text “was probably assembled under the Western Han dynasty during the first century BC.” In fact, Joseph Needham, the noted pioneer of history of Chinese science, one of Nanda’s only two references in this whole issue, mocks those who “would cheerfully put the Chou Pei 1000 years too early” and accepts a date in the Han dynasty, that is, between 206 bce and 220 ce.
This brings in an interesting aside: Nanda, as we saw, was willing to stretch Baudhāyana’s date to 200 bce, while most scholars have him earlier than 500 bce (even to “800–600 bce,” to quote the U.S. historian of Indian mathematics Kim Plofker); in contrast, Nanda curiously ages the Chinese text by at least five centuries, taking it before 600 bce — a neat somersault to suggest Chinese “priority” over the Indian text! (Of course, as pointed out by Needham and others, the Zhou Bi Suan Jing integrates older material and practices, but so do the Shulbasūtras.)
Such double standards apply to her statement that the first Indian proof, by Bhāskarāchārya in the 12th century ce, is “an ‘exact reproduction’ of the Chinese” one. This, she claims, was stated by Needham “and many others” (whom we shall not know). Actually, Bhāskara in his Bījaganita mentions two proofs which he attributes to tradition (and therefore of uncertain but older dates). One is rāshigata (arithmetical); the second, kshetragata (based on geometric algebra), does bear some likeness to the Chinese proof — but equally to the Shulbasūtra-type of constructions.
The evidence, again, is not clinching: Nanda fails to realize that likeness alone is no proof of borrowing — neither from India to China (as she blames unnamed “Indocentric historians” for always assuming) nor from China to India, as she herself favours. The methodology serious scholars follow is to note similarities and chronologies, whenever unambiguous, but to refrain from conclusions until an actual chain of transmission can be objectively established.
Finally, why should Nanda ridicule the “longstanding demand of Hinducentric historians that the theorem should be renamed ‘Baudhayana theorem’ ”? Note, once again, that she does not cite a single such historian or source to that effect; even assuming such a demand has been made, it is by no means without justification, since Baudhāyana is undeniably one of the early mathematicians to formulate the theorem (which in Greece was not formulated, let us recall, until 300 bce by Euclid). However, let us recall that mathematicians have renamed series discovered in Europe by Newton, Leibniz or Gregory as “Mādhava–Newton,” “Mādhava–Leibniz” and “Mādhava–Leibniz–Gregory” series — after Mādhava, the fourteenth-century founder of the famous Kerala School of mathematics and astronomy, who discovered the said series long before European mathematicians. Similarly, a better term for the Pythagoras theorem would have to be “Baudhāyana–Zhou Bi–Pythagoras theorem” (in whatever order). It is far too unwieldy ever to be adopted, yet would be accurate, historically justified, and certainly no insult to Pythagoras, who made a profound impact on Greek and later Western thought without leaving behind a single written work.
(To be concluded)
Note: Except for long vowels, I have made no attempt to use standard diacritics for Sanskrit words, opting instead for spellings closer to their actual pronunciation. “bce” and “ce” stand for “Before Common Era” (or BC) and “Common Era” (AD).
(Michel Danino’s main interests lie in Indian protohistory and the history of Indian science and technology; he has also authored a few papers and educational modules on the latter. He teaches at IIT Gandhinagar and is a member of ICHR. Email: firstname.lastname@example.org.)