Eternal India Encyclopedia

Ancient Concepts. Sciences & Systems

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Baudhayana in .the context of the construction of the Caturasra-

syenacit Vedic altar. For this reason, some Indian scholars call it

‘Baudhayana’s Theorem’. Baudhayana states it as follows: The

diagonal of a rectangle produces both (areas) which its length and

breadth produce separately.

Chaturastra-syenacit

altar which may have inspired the discov-

ery of the so-called Pythagorean theorem. Note that the square

on the diagonal of the interior square is exactly twice its own

area. This special case (for the square) was generalized by

Baudhayana to the rectangle. The rectangular case is equivalent

to the theorem for the right triangle that is usually found in

textbooks on geometry. So the 'Pythagorean theorem' had been

derived by Baudhauyana two thousand years before Pythagoras!

It is unclear if Baudhayana really discovered the theorem or

was simply stating a result that was already known. He does not

use the term

‘iti sruyate’’

or ‘so one hears’ in stating the result

which in fact may be constructed to mean that he was its discov-

erer, but this is speculative. It is certain however that the so-called

Pythagoras theorem was known both in theory and application

thousands of years before the time of Pythagoras. Also the idea of

the geometric proof, commonly thought to have originated with the

Greeks are found in the Sulbas in many places. They are, however,

given simply as a matter of course and not endowed with the

formalism that was to achieve such perfection at the hands of

Euclid.

The second problem of area equivalence, notably the important

special case of the conversion of squares into circles of the same

area (approximate) and vice versa also receives a good deal of

attention from the Sulba authors. Baudhayana in fact has given an

ingenious way of designing spoked chariot wheels by combining the

square on the diagonal theorem and circling the square, one of the

most interesting examples from antiquity. The square-circle prob-

lem leads to fundamental results that were used not only in India,

but also in Old-Babylonia (1700 BC) and the Egyptian Middle

Kingdom (c.2000 to 1800 BC). The Sulbas of Baudhayana and

Apasthambha derive from it the following famous unit fractions ap-

proximation for\/2.:

\/2

1+1/3 + l/(3.4) - l/(3.4.34)

"Egyptian" figures found in the Baudhyana and the Apastamba

Sulbas. These figures would have been difficult to construct using

the arithmetic methods favored by the Egyptians. Indians on the

other hand used geometric constructions, and could easily handle

such figures. This suggests that there was a good deal of contact

between India of the Sutra period and ancient Egypt._______________

The relatively late Manava Sulba gives the approximation

71=3.16049 = 4.(8/9)

used by Ahmes of Egypt (c.1550 BCE). All

this (and there is much more), as modem researchers like Seiden-

berg have pointed out, constitute overwhelming evidence for the

influence of the Sulbas on the ancient civilisations of Egypt and

West Asia. The fact that the Sulba authors derive each one of their

mathematical results from a specific religious and ritual application

has enabled Seidenberg to trace the origin of mathematics to the

Sulbas. As a result, a careful study of Vedic mathematics is

leading to a fundamental transformation of our understanding of the

history of the ancient world and not just India.

HISTORY AND CHRONOLOGY OF

VEDIC MATHEMATICS

As a result of the discovery of Seidenberg and others recognis-

ing the Sulbas as the source of the mathematics of both Old-

Babylonia (1700 BCE) and the Egyptian Middle Kingdom (c.2050

BCE), it is evident that the mathematics of the Sulbas must have

been in existence no later than 2100 BCE if not a good deal earlier.

This date is supported also by the discovery of Vedic altars whose

geometric details are found described in the Sulbas. Thus on the

basis of archaeology and Egyptian mathematics, Vedic mathemat-

ics must conservatively be placed in the middle of the third millen-

nium BCE. Historians of science have recognised the revolutionary

implications of these findings that are now being woven into mod-

em works.

To the left is the Egyptian flat-top pyramid, the

Mastaba

To the right the

Smashana-cit

(funeral altar) as described in the

Baudhayana Sulba-Sutra.

The

Mastaba

is essentially the

Samashana-cit

turned around

The prayer used is from the

Taitirya samhita

and says : "May

we gain prosperity in the world of our fathers!" This suggests

that the idea of using pyramids as resting places for the dead

may have been derived by the Egyptians from India.

This scenario, however, is in conflict with the nineteenth cen-

tury historical view that the Vedic Aryans were not present in India

before 1500 BCE and the date of 1200 BCE for the Rig Veda. But

these dates as well as the idea of the ‘Aryan invasion’ of India in

the late ancient age are now beginning to be seen to be in serious

conflict with data from archaeology, astronomy and now ancient

mathematics. New archaeological evidence, notably of the drying

up of the Saraswati river around 2000 BCE places the Rig Veda

firmly before that age. There is now an emerging consensus that

the Vedic Aryans were already established in India by 4000 BC.

Thus current archaeological data are fully in agreement with the

date for Vedic mathematics determined by comparisons with Egypt

and Old-Babylonia on one hand, and the Vedic altars found among

the Harappan ruins on the other. In any case, mathematics is

mathematics, and the evidence of mathematics now is that the

contents of the Sulbas were in existence no later than 2100 BCE.

Judging from results of the most recent research, this date if