2/N: Many supervised and unsupervised algorithms in machine learning & deep learning use distance measures today - one of which is the Euclidean distance, which is nothing but an application of the right-angled theorem to compute the difference between two n-dimensional vectors.

3/N: Right-angle triangle theorem is often ascribed to Pythagoras, an ancient mathematician from Ionia. It is believed that Pythagoras was born in 569 BC in Samos, Ionia and died 475 BC.

4/N: It is interesting to note that, unlike many later Greek mathematicians, who wrote a number of books, there are no writings by Pythagoras himself. There is no documented proof available that Pythagoras himself discovered the theorem of the right-angled triangle.

5/N: The first Greek documented mention of the right angle theorem is in the book Elements by Euclid, believed to be written around 350 BC. Specifically, Euclid's Elements Book 1: Proposition 47 mentions the theorem.

6/N: Centuries before Pythagoras was born, ancient Indians not only knew about the theorem of the right angled triangle, but used it to build very complex 3D structures. Shulba Sutras (शुल्बसूत्र) from ancient India, composed no later than 800 BC, refer to the theorem extensively

7/N: While ancient Babylonians knew of the application of the right-angled triangle, it is in ancient India where they not only analyzed theoretical aspects of the theorem in depth, but also used it to build sophisticated structures for practical usage.

8/N: Shulba Sutras are part of Kalpa Sutras in Veclic literature, an enormous body of work dealing with means of self-realization. There are four main Shulba Sutras, the Baudhayana, the Apastamba, the Manava, the Katyayana, & a number of smaller ones.

9/N: The general formats for the main Shulba Sutras are the same - each starts with sections on geometrical and arithmetical constructions and ends with details of how to build “chitis” or 2D and 3D ceremonial platforms or fire altars.

10/N: Literal meaning of the word Shulba is rope. In its true essence Shulba represents both connection & evolution from individual consciousness to the universal consciousness. Shulba Sutras are used to construct altars or 'chiti' - a basis for 'chitta' or consciousness.

11/N: When all the Shulba Sutras are viewed as a whole, a striking level of unity & elegance emerges. There are exactly the right geometrical constructions to the precise degree of accuracy necessary for the artisans to build the altars.

12/N: Nothing is redundant in Shulba Sutra collection. This point is nicely made by David Henderson who argues that the units of measurement used easily lead to the accuracy of the diagonal of one of the main bricks of "roughly one-thousandth of the thickness of a sesame seed."

13/N: Baudhyana Shulba Sutra 1:48 gives us in clear language what later came to be known to the western world as the Pythagoras Theorem: "A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together."

14/N: Katyayana Sulba Sutra 2:11, provides the same theorem of the right angled triangle in a slightly different form. “In a rectangle, it's diagonal produces by itself an area, which it's length and breath produce separately. This is the way of measuring areas”.

15/N: Katyayana Sulba Sutra 2:12, reinforces the description of the theorem using a square: “In a square a diagonal produces an area, which is double that of its side”

16/N: The same is mentioned in Baudhayana Sutra 1.45 & Apastamba Shulba Sutra

17/N: Katyayana Shulba Sutra 2:8 also provides us specific example of a triplet for right angle theorem “In a right angled triangle, whose one side is 1 pada and other side is 3 padas, the hypotenuse will produce an area of 10 square padas.”. In other words 1^2 + 3^2 = 10

18/N: Katyayana Shulba Sutra 2:9 gives us another example of the right angle triangle theorem “In a right angled triangle, whose one side is 2 padas and other side is 6 padas, the hypotenuse will be tatkarani or produce an area of 40 square padas.”. In other words 2^2 + 6^2 = 40

19/N: In Greek tradition, Pythagorean theorem is expressed in terms of geometry. Only the ancient Hindus provided a computational expression for the right angled theorem. Baudhayana Shulba Sutra 1:61, Katyayana Shulba Sutra 2:13 & Apastamba Shulba Sutra all mention this method.

20/N: The above mentioned sutras say: One should increase the measure (of the side of the square whose double producer is to be found) by its third part & again by fourth part of this third less by the thirty fourth part of this fourth. The value obtained is called savishesa

21/N: As an example, let us assume a square of 8 units on each side. According to the approximation method outlined in Shulba Sutra, the length of the diagonal L1 is given by L1 = 8 + (8/3) + ( 8 / (3 * 4)) - (8 / (3 * 4 * 34)) = 11.3137254902

22/N: If we instead apply the standard formula for finding diagonal L2 = Square-Root (8^2 + 8^2) = Square-Root (128) = 11.313708499 As you can see the computation method from Shulba Sutra (L1) is accurate up to 4 decimal points

23/N: Bhaskaracharya (Bhaskara II, no later than 1140 AD), the genius & eminent Indian mathematician, astronomer gives two different proofs of the right angle theorem.

24/N: The first one is an elegant visual proof of the right angle theorem in Siddhanta Shiromani.

25/N: In the second proof, Bhaskara-II began with a right triangle and then he drew an altitude on the hypotenuse. From here, he used the properties of similarity to prove the theorem.

26/N: Pythagoras exhibited some striking similarities in his philosophy and thought process with those prevalent in ancient India.

27/N: Pythagoras believed in the immortality of the soul, advocated vegetarianism, fasting as a way of purification, the sanctity of animal life, believed in the existence of soul in plants, animals & humans & valued the philosophy of monism.

28/N: The above mentioned beliefs were central to his school and were practiced by his followers for almost a millennium.

29/N: It is important to point out that those practices were new to Greece during the period of Pythagoras. There is only one place in the world where all these values existed prior to Pythagoras. This is India which was populated by the ancient Hindus.

30/N: Isocrates (436 - 338 BC), a Greek philosopher wrote about Pythagoras's visit to Egypt. But did Pythagoras just visit Egypt or did he visit other nations too?

31/N: There is enough literary evidence to point to the possibility that Pythagoras visited India too. Apollonius of Tyana, a well respected Pythagorean philosopher during the Roman period wrote about it.

32/N: Apollonius was revered enough that the Roman emperor Septimius Severus (146 -211 AD) erected a statue of Apollonius in his private shrine along with Christ and Abraham.

33/N: Philostratus (170 - 244 AD), a Greek sophist of the Roman imperial period, taught in Athens and then served in the court of Julia Domma in Rome, the wife of emperor Septimius Severus, and compiled the travel accounts of Apollonius.

34/N: Apollonius learned the doctrine of nonviolence in the tradition of Pythagoras who, in Apollonius’s view, was himself taught by Indian philosophers.

35/N: Philostratus quotes Apollonius: “I did not sacrifice...I don't touch blood, even on altar, since that was the doctrines of Pythagoras...it was the doctrines of the naked philosophers of Egypt & the wise men of India, from whom Pythagoras derived seeds of their philosophy”

36/N: Titus Flavius Clemens (150 - 215 AD), was a Christian theologian who was familiar with Greek philosophy. In his book “Stromata”, he writes Pythagoras went to Persia & came in contact with the Indian Brahmins. He mentioned, ”Pythagoras was a hearer of.. Brahmins”