A limited response to Meera Nanda’s essay “Hindutva’s science envy” in Frontline

This piece is an attempt at a limited response to one aspect (the “Pythagorian” matter) of Meera Nanda’s essay “Hindutva’s science envy“, published in the online edition of Frontline magazine on Aug 31, 2016.

In the above mentioned piece, Meera Nanda includes:

In what follows, three pet claims about the Indian history of science will be examined in order to sift facts from nationalist fictions. Two of these claims have to do with mathematics: the first declares Baudhayana, the ancient Indian rope-stretcher and altar-maker, to be the real discoverer of the Pythagoras theorem; the second is our most cherished “fact” that India is the birthplace of the sunya, or zero. The third claim has to do with the science of genetics and the ancient Indian understanding of heredity. (I have examined these and related themes in my recent book, Science in Saffron: Skeptical Essays in History of Science, which can be consulted for a historical and technical background for the core of the arguments presented below.)

As evidence to her hypothesis with regard to the “Pythagorean” matter, she includes:

Pythagoras, a mystic-mathematician born sometimes around 570 BCE on an island off the coast of modern-day Turkey, comes in for a lot of abuse in India. (sic) He is seen as someone who is wrongly and unfairly credited with having discovered the theorem that goes by his name, while the real discoverer was our own Baudhayana, a priest-craftsman and the composer of Baudhayana Sulvasutras, a work dated from anywhere between 800 and 200 BCE. Since Baudhayana texts predate Pythagoras, it is assumed that Pythagoras must have travelled to India and learned the theorem (along with Hindu beliefs in reincarnation and vegetarianism) from Hindu gurus. Thus, it has been a long-standing demand of Hindu-centric historians that the theorem should be renamed “Baudhayana theorem”. Not only is Baudhayana credited with the discovery of the Pythagorean theorem, he is declared to be the first to have given a proof for the theorem, the first to have calculated “Pythagorean triples”, the first to have figured out irrational numbers, the first to calculate the square root of 2 and much else. This is the sentiment that Dr Harsh Vardhan, Minister for Science and Technology, gave voice to when he spoke at the inauguration of the Science Congress last year.

All of the above claims about Baudhayana’s priority are false. They evaporate the moment one looks beyond India to see what was going on in other major civilisations around the time when the Sulvasutras were composed.

At least a millennium before Baudhayana was even born, Mesopotamians had figured out the relationship between the sides of a right-angle triangle described by Pythagoras’ theorem. Mesopotamians (and their neighbours, Egyptians) took to measuring land in order to affix the boundaries every time the Euphrates-Tigris and the Nile would flood and wash away the existing boundaries. While the Egyptian evidence comes much later, the evidence that Mesopotamians knew the theorem, had worked out Pythagorean triples, and learned to calculate the square root of 2 is etched in hardened clay tablets dating back 1800 BCE, a thousand-year lead over Baudhayana. Two clay tablets in particular—Plimpton 322 and YBC7289, housed in Columbia and Yale universities, respectively—were deciphered by Otto Neugebauer, the world’s foremost authority on the cuneiform script, in the 1940s. He and his colleagues established that while Plimpton was a table of what we today call Pythagorean triples, the Yale tablet shows a remarkably accurate calculation of the square root of 2. These tablets alone blow holes through much of the case for Baudhayana’s priority.

Moving east from Mesopotamia, east even of India, the Chinese had not only figured out the theorem but even provided a proof around the time of Confucius (approximately 600 BCE), if not earlier. The Chinese evidence comes from a text called Chou Pei Suan Ching (which translates into “The Arithmetical Classic of the Gnomon and the Circular Path of Heavens”) dated anywhere from 1100 to 600 BCE. Later, Han dynasty (third century BCE) mathematical texts formalised the theorem and named it kou-ku (or gou-gu) theorem.

The Chinese achievement brings us to the issue of proof. The Sulvasutras, being manuals for constructing altars, offer sophisticated and ingenious mathematical aids for all kinds of complex geometrical shapes and their transformations. But they do not set out to prove or justify these rules of geometry. Mesopotamians and Egyptians, too, have left no trace of a proof.

So where does the first proof of Pythagoras’ theorem come from? Whether or not Pythagoras himself offered a general proof for all right-angle triangles is not clear. The first clear proof of this theorem in the Greek tradition comes from Euclid, who lived full three centuries after Pythagoras. All evidence points to the above-mentioned Chinese text, which was written at least three centuries before Euclid, to be the first proof of Pythagoras’ theorem. Unlike Euclid’s method of logical deduction, the Chinese idea of proof was based upon producing a visual demonstration from which the general case could be inferred. The first Indian proof of this theorem comes only from Bhaskara in the 12th century, and as the eminent historian of Chinese science Joseph Needham and many others have pointed out, Bhaskara’s proof was an “exact reproduction” of the Chinese Hsuan-thu diagram from Chou Pei.

In the above excerpt, Meera Nanda, in reference to ‘the sentiment that Dr. Harsh Vardhan…gave voice to’, makes a declaration that claims about Baudhayana’s priority over Pythagoras are false.

Before getting into the actual matter of Baudhayana’s priority over Pythagoras and the nitty-gritty of the evidence she includes – Plimpton 322, YBC7289, Chou Pei Suan Ching, Shuan-thu diagram, kou-ku (or gou-gu) theorem, Euclid’s method of logical deduction – let us look at two excerpts from her own reasoning:

Excerpt 1:

Pythagoras, a mystic-mathematician born sometimes around 570 BCE on an island off the coast of modern-day Turkey, comes in for a lot of abuse in India. He is seen as someone who is wrongly and unfairly credited with having discovered the theorem that goes by his name…(sic)

Excerpt 2:

Whether or not Pythagoras himself offered a general proof for all right-angle triangles is not clear.

If, in Meera Nanda’s own words, ‘whether or not Pythagoras himself offered general proof for all right-angle triangles is not clear’, why would she not empathise with any one who sees Pythagoras ‘as someone who is wrongly and unfairly credited with having discovered the theorem’?

The words ‘general proof’, ‘all’ in Excerpt 2 seem to be carefully used, as they give her room to separate the burden of general proof from the discovery of theorem, allowing therefore for the attribution to Pythagoras to not be challenged. To see this more clearly, it is important to observe closely the points Meera Nanda seems to be making:

  1. She credits the Chinese by saying that the ‘Chinese had not only figured out the theorem but even provided a proof around the time of Confucius (approximately 600 BCE), if not earlier. The Chinese evidence comes from a text called Chou Pei Suan Ching(which translates into “The Arithmetical Classic of the Gnomon and the Circular Path of Heavens”) dated anywhere from 1100 to 600 BCE. Later, Han dynasty (third century BCE) mathematical texts formalised the theorem and named it kou-ku (or gou-gu) theorem.’
  2. She, however, makes it a point to differentiate that unlike ‘Euclid’s method of logical deduction, the Chinese idea of proof was based upon producing a visual demonstration from which the general case could be inferred.’

One way to interpret this could be that, while on the one hand, she uses the Chinese account to neutralise what she asserts (without credible verbatim evidence) as the Indian claim, in including the ‘logical deduction’ aspect of Euclid’s method, she allows for the Greek (Pythagoras + Euclid) antiquity to not be challenged (read: dropping of attribution to Pythagoras), for instance by the Chinese. Note also, her comfort in not calling out the fact that the Chinese have their own name for the theorem (and, imho, rightly so) which is based on their indigenous history while she seems clearly uneasy with an Indian move to rename it as per their indigenous history.

Let us now look at some of the data she includes with regard to the “Pythagorian” matter.

Chronological elements, with regard to the “Pythagorian” matter, included in Meera Nanda’s essay. The author is re-using them, only to point out inconsistencies and implications of Meera Nanda’s data. The author does not uphold any of these dates.

Century BCE

Chinese

Greek

Indian

11

Chou Pei Suan Ching

   

10

9

8

Period of Shulba Sutras, as per Meera Nanda. (She uses ‘Sulva Sutras’ in her article)

7

6

Pythagoras

5

   

4

3

Kou-ku theorem

Euclid

2

   

If not needing to provide general proof and only discovering the theorem is sufficient, should Nanda still not empathise with any voice that calls for reconsideration of a global attribution to Pythagoras, given both the Chinese and Indian claims are still in contention, even going by just her own data and the criteria of just theorem, no proof?

Coming now to the matter of an Indian claim and the dating of Sulba Sutras, let us look at an excerpt from the book “Science in India – A historical perspective” from an authentic and undoubtedly qualified Indian voice, Dr. B.V. Subbarayappa:

In the history of mathematics, the Pythagorean theorem has no doubt an honoured position. But whether the Pre-Socratic Greek philosopher-mathematician, Pythagoras (c. sixth century BCE) was the actual propounder of this theorem is still an unsolved question. He was no doubt an outstanding Pre-Socratic thinker who even thought that numbers were the primordial attributes of all things, and 10 was the perfect one which comprised the whole of numbers. His name as associated with this theorem appears in the much later works of Cicero (first century BCE), Diaogenes Laettrius (second century BCE), Heron (third century BCE), and others. A knowledge of the type expounded in the Pythagorean theorem could also be found among the Egyptians and the Chinese, although it is not so well delineated as in the Indian Sulba sutras.

In the history of science, it is essential to understand the idea of thought-structure in its context. The so-called Pythagorean theorem arose out of the general idea of Pythagoreans. In number they saw many things numerically—even reason and justice. They regarded the whole heaven as a musical and numerical scale, and conceived of the ‘harmony of the spheres’. On the other hand, Sulba sutras presented the theorem in terms of the square on a diagonal (in a rectangle) being equal to the sum of the squares of the two sides of the right-angled triangle thus formed. Each of these two approaches needs to be understood in its context to appreciate its originality. (p. 239-40)

Amongst other things, B.V. Subbarayappa’s objectivity should clearly stand out, in light of the text in bold in the excerpt above.

Note that this credible Indian voice:

  • Despite  the point that (and to use Meera Nanda’s words) ‘whether or not Pythagoras himself offered a general proof for all right-angle triangles is not clear’, not only appreciates Pythagoras (Bolded text 1) but also calls for his approach ‘to be understood in its context to appreciate its originality’ (Bolded text 4),
  • and clearly acknowledges the Egyptian and Chinese in saying that ‘A knowledge of the type expounded in the Pythagorean theorem could also be found among the Egyptians and the Chinese’.

Before getting to the  “Insider” (to use Rajiv Malhotra’s term) dating of Shulba Sutra, let us first baseline how “well delineated” the Indian Sulba theorem is, included in p. 37 (included below) of the book “Pride of India“:

2016-09-01 12.09.20

While one could refer to related scholarship of many people including the likes of A. Seidenberg, Frits Staal, David Bailey, Subhash Kak, Radha Charan Gupta, Vinod Mishra, B Datta, S G Dani, A K Dutta, George Gheverghese Joseph, John Price, A. N. Singh, A. K. Bag, Kim Plofker, Georges Ifrah, R. N. Iyenger, for the limited scope of this piece, it might suffice to refer to one scholarly treatment with regard to Sulba Sutras that can be found in Amartya Kumar Dutta’s “Mathematics in Ancient India“, where, amongst other things, the following (in bold) is included as regards the dating of the Sulba Sutras:

The oldest known mathematics texts in existence are the Sulba-sutras of Baudhayana, Apastamba and Katyayana which form part of the literature of the Sutra period of the later Vedic age. The Sulbasutras had been estimated to have been composed around 800 BC (some recent researchers are suggesting earlier dates). But the mathematical knowledge recorded in these sutras (aphorisms) are much more ancient; for the Sulba authors emphasise that they were merely stating facts already known the composers of the Brahmanas and Samhitas of the early Vedic age. ([3], [1], [2]).

[…]

Constructions of the fire-altar are described in an enormously developed form in the Satapatha Brahmana (c. 2000 BC; vide [3]); some of them are mentioned in the earlier Taittiriya Samhita (c. 3000 BC; vide [3]); but the sacrificial fire-altars are referred – without explicit construction – in the even earlier Rig Vedic Samhitas, the oldest strata of the extant Vedic literature.

Plane geometry stands on two important pillars having applications throughout history: (1) the result popularly known as the ‘Pythagoras theorem’ and (ii) the properties of similar figures. In the Sulbasutras, we see an explicit use of the Pythagoras theorem and its applications in various geometric constructions such as construction of square equal (in area) to the sum, or difference, of two given squares, or to a rectangle, or to the sum of n squares. These constructions implicitly involve application of algebraic identities such as […]. These reflect a blending of geometric and subtle algebraic thinking and insight which we associate with Euclid. In fact, the Sulba construction of a square equal in area to a given rectangle is exactly the same as given by Euclid several centuries later ! There are geometric solutions to what are algebraic and number-theoritic problems.

Pythagoras theorem was known in other ancient civilisations like the Babylonian, but the emphasis there was on the numerical and not so much on the proper geometric aspect while in the Sulbasutras one sees depth in both aspects – especially the geometric. This is a subtle point analysed in detail by Seidenberg. From certain diagrams described in the Sulbasutras, several historians and mathematicians like Burk, Hankel, Schopenhauer, Seidenberg and Van der Waerden have concluded that the Sulba authors possessed proof of geometrical results including the Pythagoras theorem – some of the details are analysed in the pioneering work of Datta ([2]). One of the proofs of the Pythagoras theorem, easily deducible from the Sulba verses, is later described more explicitly by Bhaskara II (1150 AD).

In context of the  seeming centrality Meera Nanda ascribes to the logicality of Euclid’s proof, should not Dutta’s point that ‘In fact, the Sulba construction of a square equal in area to a given rectangle is exactly the same as given by Euclid several centuries later!’ require explicit reconciliation by Nanda given the Sutra clearly precedes Euclid even by her own dating data (atleast for the most part), unless of course she makes the almost-laughable claim of a dating for the Sutra after Euclid?

From the above, should it not follow, to any rational mind, that “ascribing to Pythagoras” is not only not globally meaningful (certainly atleast not objective to Chinese and Indian sensibilities), but is also suspect, if the purpose of ascribing is to denote priority in discovery?

Meera Nanda might perhaps benefit in reading what Kim Plofker notes, in ancient Indian mathematics, “True perception, reasoning, and authority were expected to harmonize with one another, and each had a part in supporting the truth of mathematics.” [19, pg. 12].

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