BAUDHAYANA (PYTHAGORAS) THEOREM (World Guru of Mathematics, Part – 9)

Posted on August 31, 2011

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BAUDHAYANA (PYTHAGORAS) THEOREM

                                    It was ancient Indians mathematicians who discovered Pythagoras theorem. This might come as a surprise to many, but it’s true that Pythagoras theorem was known much before Pythagoras and it was Indians who actually discovered it at least 1000 years before Pythagoras was born!

Baudhayana

It was Baudhāyana who discovered the Pythagoras theorem. Baudhāyana listed Pythagoras theorem in his book called Baudhāyana Śulbasûtra (800 BCE). Incidentally, Baudhāyana Śulbasûtra is also one of the oldest books on advanced Mathematics. The actual shloka (verse) in Baudhāyana Śulbasûtra that describes Pythagoras theorem is given below :

“dīrghasyākaayā rajjuH pārśvamānī, tiryaDaM mānī, cha yatpthagbhUte kurutastadubhayā karoti.”

Interestingly, Baudhāyana used a rope as an example in the above shloka which can be translated as – A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together. As you see, it becomes clear that this is perhaps the most intuitive way of understanding and visualizing Pythagoras theorem (and geometry in general) and Baudhāyana seems to have simplified the process of learning by encapsulating the mathematical result in a simple shloka in a layman’s language.

Some people might say that this is not really an actual mathematical proof of Pythagoras theorem though and it is possible that Pythagoras provided that missing proof. But if we look in the same Śulbasûtra, we find that the proof of Pythagoras theorem has been provided by both Baudhāyana and Āpastamba in the Sulba Sutras! To elaborate, the shloka is to be translated as -

The diagonal of a rectangle produces by itself both (the areas) produced separately by its two sides.

Modern Pythagorean Theorem

The implications of the above statement are profound because it is directly translated into Pythagorean Theorem (and graphically represented in the picutre on the left) and it becomes evident that Baudhāyana proved Pythagoras theorem. Since most of the later proofs (presented by Euclid and others) are geometrical in nature, the Sulba Sutra’s numerical proof was unfortunately ignored. Though, Baudhāyana was not the only Indian mathematician to have provided Pythagorean triplets and proof. Āpastamba also provided the proof for Pythagoras theorem, which again is numerical in nature but again unfortunately this vital contribution has been ignored and Pythagoras was wrongly credited by Cicero and early Greek mathematicians for this theorem. Baudhāyana also presented geometrical proof using isosceles triangles so, to be more accurate, we attribute the geometrical proof to Baudhāyana and numerical (using number theory and area computation) proof to Āpastamba. Also, another ancient Indian mathematician called Bhaskara later provided a unique geometrical proof as well as numerical which is known for the fact that it’s truly generalized and works for all sorts of triangles and is not incongruent (not just isosceles as in some older proofs).

One thing that is really interesting is that Pythagoras was not credited for this theorem till at least three centuries after! It was much later when Cicero and other Greek philosophers/mathematicians/historians decided to tell the world that it was Pythagoras that came up with this theorem! How utterly ridiculous! In fact, later on many historians have tried to prove the relation between Pythagoras theorem and Pythagoras but have failed miserably. In fact, the only relation that the historians have been able to trace it to is with Euclid, who again came many centuries after Pythagoras!

Bhaskara's Proof

This fact itself means that they just wanted to use some of their own to name this theorem after and discredit the much ancient Indian mathematicians without whose contribution it could’ve been impossible to create the very basis of algebra and geometry!

                     Many historians have also presented evidence for the fact that Pythagoras actually travelled to Egypt and then India and learned many important mathematical theories (including Pythagoras theorem) that western world didn’t know of back then! So, it’s very much possible that Pythagoras learned this theorem during his visit to India but hid his source of knowledge he went back to Greece! This would also partially explain why Greeks were so reserved in crediting Pythagoras with this theorem!

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Posted in: Mathematics