Posted on: 22 January 2015

Research Paper:

Pythagorean Triples and Cryptographic Coding

By Subhash Kak

Oklahoma State University, Stillwater - 2010

"The earliest statement of the theorem of the square on the diagonal (Pythagoras theorem), together with some examples, is to be found in the geometry text of Baudh?yana (c. 800 BC). "

A Pythagorean triple (a, b, c) consists of positive integers that are the sides of a right triangle and thus a2 + b2 = c2. Given a Pythagorean triple (a, b, c), we have other similar triples that are d(a, b, c), where d > 1. A primitive Pythagorean triple (PPT) consists of numbers that are relatively prime.

Pythagorean triples have been found on cuneiform tablets of Babylon and they are important in Vedic ritual and described in early geometry books of India and in the works of Euclid and Diophantus.

The earliest statement of the theorem of the square on the diagonal (Pythagoras theorem), together with some examples, is to be found in the geometry text of Baudh?yana (c. 800 BC). In his ?ulba S?tra 1.12 and 1.13, it is stated:

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, and 15 and 36.

O’Conner and Robertson in their history of mathematics project take Baudh?yana to be 800 BC. Seidenberg presents various arguments for an early date for the knowledge of the Pythagoras theorem in India (for other aspects of early Indian mathematics, and for the altar ritual that provides the context in which this mathematics was used in the ?ulba S?tras, see. Van der Waerden saw a ritual origin to the discovery of Pythagoras.

The ?ulba S?tras are texts of applied geometry that provide techniques to draw altars of different shapes and sizes in a convenient manner. The word ?ulba means a “cord”, “rope”, or “string” and the root ?ulb signifies “measurement”. The cord has marks (nyańcana in Sanskrit) that indicate where the intermediate pegs are to be fixed. Thus a cord of 12 units length with nyańcana at 3 and 7, can be readily stretched to yield the right-angled triangle (3,4,5).

Read more:

http://bit.ly/1AU8Cri

nice.....

good

It was also stated in ICWA Study material.

Dutt & Singh said it was the very first

But won't the sickulars accuse you of flying planes in 'pagan' India? Does not all knowledge flow out of the gun of the Christian West? Were not the tufted Brahmins and the intellectuals of the Indian past nothing but obscurantist idiots? The convent educated deracinated kind is going to b up in arms!

I think all that is said is that a rope marked in the proportion of 3:4:5 produces a right triangle when wound around pegs fixed at any two markings. It should not be difficult to find the primitive Pythogorean triplets (PPTs) using any length of a rope through trial and error, containing pairs that are relatively prime..

This is the ours Ashian history. Unfortunately most of the people don't accept.

If there were no such science and mathematics in ancient India, there would not have been a surge of architectural development, especially in the north eastern part of the Subcontinent. Without the concepts of unit measurements, basic arithmatical and geometrical proofs, and so on, it's impossible to build citadels we find in the Harappan sites. The above geometrical concept was of much later period. Indian and Arabic mathematical advancements were of course one of the earliest in the world. even more intricate mathematical approaches are found in Gupta Era around tenth century AD. What we need is to determine the stages of these advancements and their approximate datings.

- The guy sure knew the ropes!

The derision with which modern Indian pseudo- secularists discard with scorn the revelations of our ancient texts, shows the deep rooted inferiority complex of Indians, inculcated by centuries of foreign rule by forest the barbaric Muslims & then by the ' cultured' Europeans.

@sabyasachi chatterjee bedouin of arabia contributed nothing this world...those mathematicians were persian aka ex zoroastrianism...they were translating sanskrit,greek and chinese books into persian and arabic...

http://www.storyofmathematics.com/sumerian.html

http://www-history.mcs.st-and.ac.uk/Indexes/Indians.html

Narad Sutras clearly propound Pythagoras theorem, much before the west came to know of it. And intricate geometry of ancient temples, which stand, head held high even after thousand or more years, is a proof enough of the applied knowledge. Temples that submerged in Bhakra dam's lake area 50 years ago, still raise their head when level in Gobind Sagar lake is low. These can be easily seen from the National Highway , just few Kilometers short of Bilaspur, when level is low at a particular time of the year.

Excellent link! Thank you!

Lilabati's note is also noticeable in this regard

Very good. But why the sloka was not translated word to word ? That would have given us the extended knowledge of what Samskruta word denotes the diagonal and other sides. Now I have to run to many sites to get it.

Sunil Paii

Highly obliged

Sabyasachi Chatterjee Indian and Arabic mathematical advancements were of course one of the earliest in the world Please read the attached notes carefully.. :)

Arabs were zero not only in mathematics but in all the other knowledges ...

Excellent! Can you mention the source and title of this page?

Book name : India's contribution to World Thought and Culture Research paper title : India's contributions to Arab Civilisation Author : W.H. Siddiqi .. Page nos. 585, 586 & 587

But see what we are taught today.. !! :( even the educated pro-Indians peoples like Sabyasachi Chatterjee thinks that we owe a lot to Arabs even in mathematics.. :(

only indians say that !

great

Probably Pythogoras read this before proposing his famous theorem