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.

The relatively late Manava Sulba gives the approximation 71=3.16049 = 4.(8/9) 2 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. 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. HISTORY AND CHRONOLOGY OF VEDIC MATHEMATICS 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

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._______________