BAUDHAYANA THEOREM now known to be as Pythagoras Theorem

Baudhayana

                     Baudhayana

BAUDHAYANA (PYTHAGORAS)  THEOREM :

BAUDHAYANA (PYTHAGORAS) THEOREM (World Guru of Mathematics) – This is intended to spotlight the facts about India, that have been erased from history. India has been rich in culture, religion, tradition, science, literature and prosperity since ever it was found to be existed and before as well. We have been grown up without knowing our prosperity, which is a must. Indians should not lose their pride, uniqueness and whatever. Here are some facts which I came to know, that are masked from the world history.

 

The Shulba Sutras are part of the larger corpus of texts called the Shrauta Sutras, considered to be appendices to theVedas. They are the only sources of knowledge of Indian mathematics from the Vedic period. Unique fire-altar shapes were associated with unique gifts from the Gods. For instance, “he who desires heaven is to construct a fire-altar in the form of a falcon”; “a fire-altar in the form of a tortoise is to be constructed by one desiring to win the world of Brahman” and “those who wish to destroy existing and future enemies should construct a fire-altar in the form of a rhombus”.

The four major Shulba Sutras, which are mathematically the most significant, are those composed by Baudhayana,Manava, Apastamba and Katyayana, about whom very little is known. The texts are dated by comparing their grammar and vocabulary with that of other Vedic texts. The texts have been dated from around 800 BCE to 200 CE, with the oldest being a sutra attributed to Baudhayana around 800 BCE to 600 BCE.

There are competing theories about the origins of the geometrical material found in the Shulba sutras. According to the theory of the ritual origins of geometry, different shapes symbolized different religious ideas, and the need to manipulate these shapes led to the creation of the pertinent mathematics. Another theory is that the mystical properties of numbers and geometry were considered spiritually powerful and consequently, led to their incorporation into religious texts.

 

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 :

“dīrghasyākṣaṇayā rajjuH pārśvamānī, tiryaDaM mānī, cha yatpṛthagbhUte kurutastadubhayāṅ 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!

 

The later Sulba-sutras represent the ‘traditional’ material along with further related elaboration of Vedic mathematics. The Sulba-sutras have been dated from around 800-200 BC, and further to the expansion of topics in the Vedangas, contain a number of significant developments.
These include first ‘use’ of irrational numbers, quadratic equations of the form a x2 = c and ax2 + bx = c, unarguable evidence of the use of Pythagoras theorem and Pythagorean triples, predatingPythagoras (c 572 – 497 BC), and evidence of a number of geometrical proofs. This is of great interest as proof is a concept thought to be completely lacking in Indian mathematics.

Example 4.2.1: Pythagoras theorem and Pythagorean triples, as found in the Sulba Sutras.

The rope stretched along the length of the diagonal of a rectangle makes an area which the, vertical and horizontal sides make together.
In other words:

a2 = b2 + c2

Examples of Pythagorean triples given as the sides of right angled triangles:

5, 12, 13
8, 15, 17
12, 16, 20
12, 35, 37

Of the Sulvas so far ‘uncovered’ the four major and most mathematically significant are those composed by Baudhayana, Manava, Apastamba and Katyayana (perhaps least ‘important’ of the Sutras, by the time it was composed the Vedic religion was becoming less predominant). However in a paper written 20 years ago S Sinha claims that there are a further three Sutras, ‘composed’ by Maitrayana, Varaha and Vadhula (SS1, P 76). I have yet to come across any other references to these three ‘extra’ sutras. These men were not mathematicians in the modern sense but they are significant none the less in that they were the first mentioned ‘individual’ composers. E Robertson and J O’Connor have suggested that they were Vedic priests (and skilled craftsmen).

It is thought that the Sulvas were intended to supplement the Kalpa (the sixth Vedanga), and their primary content remained instructions for the construction of sacrificial altars. The name Sulvasutra means ‘rule of chords’ which is another name for geometry.

N Dwary states:

…They offer a wealth of geometrical as well as arithmetical results.

R Gupta similarly claims:

…The Sulba-sutras are (quite) rich in mathematical contents.

With reference to the possible appearance of proof is a quote from A Michaels:

...Vedic geometry, though non-axiomatic in character, is provable and indeed proof is implicit in several constructions             prescribed in the Sulba-sutras.

This is not particularly compelling evidence but does suggest that the composers of the sulba-sutras may have had a greater depth of knowledge than is generally thought.

Many suggestions for the value of p are found within the sutras. They cover a surprisingly wide range of values, from 2.99 to 3.2022.

Pythagoras’s theorem and Pythagorean triples arose as the result of geometric rules. It is first found in the Baudhayana sutra – so was hence known from around 800 BC. It is also implied in the later work of Apastamba, and Pythagorean triples are found in his rules for altar construction. Altar construction also led to the discovery of irrational numbers, a remarkable estimation of 2 in found in three of the sutras. The method for approximating the value of 2 gives the following result:

2 = 1 + 1/3 + 1/3.4 – 1/3.4.34

This is equal to 1.412156…, which is correct to 5 decimal places.

It has been argued by scholars seemingly attempting to deprive Indian mathematics of due credit, that Indians believed that 2 = 1 + 1/3 + 1/3.4 – 1/3.4.34 exactly, which would not indicate knowledge of the concept of irrationality. Elsewhere in Indian works however it is stated that various square root values cannot be exactly determined, which strongly suggests an initial knowledge of irrationality.

Indeed an early method for calculating square roots can be found in some Sutras, the method involves repeated application of the formula: A = (a2 + r) = a + r/2ar being small.

Square roots

Altar construction also led to an estimation of the square root of 2 as found in three of the sutras. In the Baudhayana sutra it appears as:

2.12. The measure is to be increased by its third and this [third] again by its own fourth less the thirty-fourth part [of that fourth]; this is [the value of] the diagonal of a square [whose side is the measure].

which leads to the value of the square root of two as being:

One conjecture about how such an approximation was obtained is that it was taken by the formula:

with  and

which is an approximation that follows a rule given by the twelfth century Muslim mathematician Al-Hassar. The result is correct to 5 decimal places.

This formula is also similar in structure to the formula found on a Mesopotamian tablet from the Old Babylonian period (1900-1600 BCE).

which expresses  in the sexagesimal system, and which too is accurate up to 5 decimal places (after rounding).

Indeed an early method for calculating square roots can be found in some Sutras, the method involves the recursive formula:  for large values of x, which bases itself on the non-recursive identity  for values of r extremely small relative to a.

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