This article is in continuation of my articles on history of Mathematics. In my earlier articles, I had presented an overview of History of Mathematics and how ancient references and works of Indians are ignored while those that came later were recognised; List of Mathematicians before Christ; Twenty Nine Sutras of Mathematics; Concept of Infinity and Nothing.In the present article let me explore more, especially advanced Mathematical Concepts in Vedas.

For the Twenty Nine Sutras of Vedic Mathematics, criticism is that it is not actually found in the Vedic Texts and it is only ‘Tricks of Vedic Thought ‘ The system works and nobody has refuted it.So if these Sutras were tricks and how come they work? If it is misinterpretation, then the Inventor should have been awarded the Nobel Prize for Mathematics!

Before attempting to understand Indian Concepts, including metaphysics, Science,the student should remember that Vedas are Spiritual in Nature; they give importance to Ontology and Self Realisation and what we call as Scientific knowledge which is useful for day to day life is regarded as lower knowledge,Apara Vidya.The practical applications of Scientific thoughts can be found in Vedic Practices,like designing a an Altar for Yagnya; the velocity of Light in the description of Surya: the movement of the Sun in the Galaxy and Universe in Sisumara Chakra; the Milky way galaxy in Vishnu’s description; Mitochondrial Pairs on Chamaka; Astronomy and Cosmology in Purana;Large numbers,Very small numbers in Astronomical texts of India;Of Nothing and Infinity in Nadadiya Suktha.And there is more.So one should read carefully,nay they are to be studied,not Read.And Vedas have sub texts or Limbs. They are called Vedangas. These contain information that is needed to lead life. These texts also contain scientific truths.

Let us look at some Vedic Concepts in Mathematics. I have quoted from sources like Wikipedia and the sources for Wiki articles.Wiki sources have been verified by me and am providing references in this article.

Several Mathematicians and Historians mention that the earliest of the texts were written beginning in 800 BCE by Vedic Hindus based on compilations of an oral tradition dating back to 2000 BCE.

Datta, Bibhutibhusan (1931). “On the Origin of the Hindu Terms for “Root””. The American Mathematical Monthly. 38 (7): 371–376. doi:10.2307/2300909. ISSN 0002-9890. JSTOR 2300909.

The Vedic civilization originated in India bears the literary evidence of Indian culture, literature, astronomy and mathematics. Written in Vedic Sanskrit the Vedic works, Vedas and Vedangas (and later Sulbasutras) are primarily religious in content, but embody a large amount of astronomical knowledge and hence a significant knowledge of mathematics. Some chronological confusion exists with regards to the appearance of the Vedic civilization. S Kak states in a very recent work that the time period for the Vedic religion stretches back potentially as far as 8000BC and definitely 4000BC. It is also worthwhile briefly noting the astronomy of the Vedic period which, given very basic measuring devices (in many cases just the naked eye), gave surprisingly accurate values for various astronomical quantities. These include the relative size of the planets the distance of the earth from the sun, the length of the day, and the length of the year. Some of Vedic works are:

- All four arithmetical operators (addition, subtraction, multiplication and division).
- A definite system for denoting any number up to 1055 and existence of zero.
- Prime numbers.

Among the other works mentioned, mathematical material of considerable interest is found:

*Arithmetical sequences, the decreasing sequence 99, 88, … , 11 is found in the Atharva-Veda*.- Pythagoras’s theorem, geometric, constructional, algebraic and computational aspects known. A rule found in the
*Satapatha Brahmana*gives a rule, which implies knowledge of the Pythagorean theorem, and similar implications are found in the Taittiriya Samhita. - Fractions, found in one (or more) of the
*Samhitas*. - Equations, 972
*x*^{2}= 972 +*m*for example, found in one of the*Samhitas*. Sites Google

**Sulba Sutras**

The

orShulba Sutras( rope”) are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction.Śulbasūtras

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 attributed to Baudhayana, Manava, Apastamba and Katyayana.Their language is late Vedic Sanskrit, pointing to a composition roughly during the 1st millennium BCE. The oldest is the sutra attributed to Baudhayana, possibly compiled around 800 BCE to 500 BCE. Pingree says that the Apastamba is likely the next oldest; he places the Katyayana and the Manava third and fourth chronologically, on the basis of apparent borrowings. According to Plofker, the Katyayana was composed after “the great grammatical codification of Sanskrit by Pāṇini in probably the mid-fourth century BCE”, but she places the Manava in the same period as the Baudhayana…There are multiple commentaries for each of the Shulba Sutras, but these were written long after the original works. The commentary of Sundararāja on the Apastamba, for example, comes from the late 15th century CE and the commentary of Dvārakãnātha on the Baudhayana appears to borrow from Sundararāja.According to Staal, certain aspects of the tradition described in the Shulba Sutras would have been “transmitted orally”, and he points to places in southern India where the fire-altar ritual is still practiced and an oral tradition preserved. The fire-altar tradition largely died out in India, however, and Plofker warns that those pockets where the practice remains may reflect a later Vedic revival rather than an unbroken tradition. Archaeological evidence of the altar constructions described in the Shulba Sutras is sparse. A large falcon-shaped fire altar (*śyenaciti*), dating to the second century BCE, was found in the excavations by G. R. Sharma at Kausambi, but this altar does not conform to the dimensions prescribed by the Shulba Sutras..

References and citations. https://en.m.wikipedia.org/wiki/Shulba_Sutras

- Plofker (2007), p. 387, “Certain shapes and sizes of fire-altars were associated with particular gifts that the sacrificer desired from the gods: ‘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’; ‘those who wish to destroy existing and future enemies should construct a fire-altar in the form of a rhombus’ [Sen and Bag 1983, 86, 98, 111].”
- ^
^{a}^{b}^{c}Plofker (2007), p. 387 - ^
^{a}^{b}Pingree (1981), p. 4 - ^
^{a}^{b}Plofker (2009), p.18 **^**Plofker (2009), p. 11**^**Pingree (1981), p. 6**^**Delire (2009), p. 50**^**Staal (1999), p. 111**^**Plofker (2009), p 19.**^**Bürk (1901), p. 554**^**Heath (1925), p. 362**^**“Square Roots of Sulbha Sutras”.*pi.math.cornell.edu*. Retrieved 2020-05-24.**^**Datta, Bibhutibhusan (1931). “On the Origin of the Hindu Terms for “Root””.*The American Mathematical Monthly*.**38**(7): 371–376. doi:10.2307/2300909. ISSN 0002-9890. JSTOR 2300909.**^**Gupta (1997), p. 154**^**Staal (1999), pp. 106, 109–110**^**Seidenberg (1978)**^**van der Waerden (1983)**^**Van der Waerden, Barten L (1983).*Geometry and Algebra in Ancient Civilizations*. Springer Verlag. p. 12. ISBN 0387121595.**^**Joseph, George Gheverghese (1997). “What Is a Square Root? A Study of Geometrical Representation in Different Mathematical Traditions”.*Mathematics in School*.**26**(3): 4–9. ISSN 0305-7259. JSTOR 30215281.**^**Boyer (1991), p. 207, “We find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. However all of these triads are easily derived from the old Babylonian rule; hence, Mesopotamian influence in the*Sulvasutras*is not unlikely. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides, but this form of the Pythagorean theorem also may have been derived from Mesopotamia. … So conjectural are the origin and period of the*Sulbasutras*that we cannot tell whether or not the rules are related to early Egyptian surveying or to the later Greek problem of altar doubling. They are variously dated within an interval of almost a thousand years stretching from the eighth century B.C. to the second century of our era.”**^**Krishnan, K S (2019).*Origin of Vedas, Chapter 5*. Notion Press. ISBN 978-1645879800.**^**Seidenberg (1983), p. 121**^**Pingree (1981), p. 5**^**Plofker (2009), p. 17**^**Thibaut (1875), pp. 232–238**^**Plofker (2007), pp. 388–389**^**Boyer (1991), p. 207**^**Joseph, G.G. (2000).*The**Crest of the Peacock: The Non-European Roots of Mathematics*. Princeton University Press. p. 229. ISBN**^**Thibaut (1875), pp. 243–246- ^
^{a}^{b}Plofker (2007), pp. 388-391 - ^
^{a}^{b}^{c}Plofker (2007), p. 391 **^**Plofker (2007), p. 392,

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