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