Pythagorean geometry in Vedic-era texts

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Controversy over the theorem-

• As per Karnataka government paper, the Pythagoras theorem is disputed in many international forums. Not the content, but Pythagoras claiming it as his own.
• It added that there are theories that say there was nobody called Pythagoras.
• The paper referred to a text called the Baudhayana Sulbasutra, in which a specific shloka refers to the theorem.

Pythagoras & the theorem

• Some evidence suggests that Pythagoras,  the Greek philosopher  (around 570–490 BC) did exist. 
• However, there is an element of mystery around him, largely because of the secretive nature of the school/society he founded in Italy. 
• Relatively little is known about his mathematical achievements, because there is nothing today of his own writings (History of Mathematics Archive, University of St Andrews, Scotland).
• The Pythagoras theorem describes the relationship connecting the three sides of a right triangle (one in which one of the angles is 90°): a² + b² = c² where a and b are the two perpendicular sides, and c is the length of the diagonal side.
• If any two sides of a right triangle are known, the theorem allows you to calculate the third side. 
• Extended to the sides of squares and rectangles and their diagonals, the equation is of immense importance in construction, navigation and astronomy.

How did knowledge of the equation evolve?

• The earliest evidence is from the Old Babylonian civilisation (1900-1600 BCE). 
• They were well aware of Pythagoras’s theorem: but since that was more than a thousand years before Pythagoras, they did not call it that: they referred to it as the ‘Diagonal rule’. 
• There are probably at least half a dozen tablets that establish that they knew this rule. The earliest evidence of a proof comes from a period after the sulbasutras. 
• The oldest surviving axiomatic proof of the theorem is in the Elements of Euclid from around 300 BCE. 
• But again, it seems extremely likely that many more-or-less formal geometric rationales for the ‘Pythagorean’ relation were understood by many of its users long before Euclid recorded his rigorous-demonstration version.

The theorem in Indian History-

• There are references in the sulbasutras, which are texts pertaining to fire rituals (yajanas) performed by Vedic Indians. The oldest of these is the Baudhayana Sulbasutra.
• The period of Baudhayana Sulbasutra is uncertain (as with other sulbasutras), there being no direct internal evidence useful in this respect. 
• By and large, in recent literature, Baudhayana Sulbasutra is taken to be from around 800 BCE .
• It has been known for quite long in academic circles that Baudhayana Sulbasutra contains a statement of what is called Pythagoras theorem (it was known rather as a geometric fact, and not as a ‘theorem’).
• The yajna rituals involved construction of altars (vedi) and fireplaces (agni) in a variety of shapes such as isosceles triangles, symmetric trapezia, and rectangles. The sulbasutras describe steps towards construction of these figures with prescribed sizes.
• The Pythagorean equation comes into play in these procedures, which involve drawing perpendiculars.  
• However, there is no evidence that the Indians had a proof, nor even that Pythagoras did.
• The idea of a mathematical proof based on an axiomatic structure is unique to the Greeks — thus in respect of the other cultures, ‘proof’ of a geometrical statement only meant some means of carrying conviction, which would have been there in various cultures. 
•  An Associate Professor from New York, who specialises in the history of Indian mathematics, cited two of the sutras in the first chapter in the Baudhayana Sulbasutra: “The areas [of the squares] produced separately by the length and the breadth of a rectangle together equal the area [of the square] produced by the diagonal. This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36.”

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