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Tag: History of mathematics

  • Numbers Arithmetic Geometric Sequences In Vedas

    When one talks about the History of Mathematics,one must also remember to talk about Numbers and their origin. Numbers, in general ,have been classified into three types.
    • Decimal
    • Unary
    • Positional
    • Binary.

    I have provided information towards the close of the post the details with Links. Now let’s look at the Vedic connection of Numbers.

    Numerals in Vedas image.
    Numbers in Sanskrit, Vedas .

    One shall find references to Numbers in the Vedas. The decimal system and expressing numbers in terms of Tens is mentioned in the Vedas. For instance, the Chamakam Anuvaka refers to Numbers.

    Worshippers of Lord Siva recite Rudram with 11 sections followed by Chamakam with 11 sections as a routine prayer every day. This is called the daily nyasam or mode of worship. In the Rudram part, the devotee pays repeated obeisance to Lord Siva and prays for his blessings for human well being. But on special occasions, the number of times the recitation is done is increased.

    In Rudra Ekadasi, Rudram is recited 11 times and Chamakam is recited once. After Rudram is recited once, one section or anuvaka ofChamakam is recited in order.

    In Laghurudram, Rudra Ekadasi is done 11 times, that is, Rudram is recited 112  or 121 times and Chamakam is recited 11 times.

     In Maharudram, 11 Laghurudrams are recited; that is, Rudram is recited 113 = 1331 times and Chamakam 112 = 121 times.

    In Atirudram, 11 Maharudrams are recited; that is, Rudram is recited 114 = 14641 times and Chamakam is recited 113 = 1331 times.

    The Chamakam mentions completely the ideal of human happiness and defines in the highest degree the desires to be fulfilled without delimiting those to be asked for or to be granted..

    DNA and Mathematics in Sri Rudram.

    In the Chamakam, in anuvakas or sections 1 to10, the devotee prays for almost everything needed for human happiness and specifies each item.  But in the 11th anuvaka or 11th section of Chamakam, the devotee prays for the desired things not specifically but in terms of numbers, first in terms of odd numbers from 1 to 33 and later in multiples of 4 from 4 to 48, as follows:

    “Eka cha me, thisrascha may, pancha cha may, sapta cha may, Ekadasa cha may, trayodasa cha may, panchadasa cha may, saptadasa cha may, Navadasa cha may, ek trimshatis cha may, trayovimshatis cha may, Panchavimshatis cha may, saptavimshatis cha may, navavimshatis cha may, Ekatrimshatis cha may, trayatrimshatis cha may, panchatrimshatis cha may, Chatasras cha may, ashtou cha may, dwadasa cha may, shodasa cha may, Vimsatis cha may, chaturvimshatis cha may, ashtavimshatis cha may, Dwathrimashatis cha may, shatstrimshas cha may, chatvarimshas cha may, Chatuschatvarimshas cha may, ashtachatvarimshas cha may”
    which means:

    “Let these be granted to me. One, three, five, seven, nine, eleven, thirteen, seventeen, nineteen, twenty one, twenty three, twenty five, twenty seven, twenty nine, thirty one and thirty three as also four, eight, twelve, sixteen, twenty, twenty four, twenty eight, thirty two, thirty six, forty, forty four and forty eight”.

    Traditional scholars and pandits explain the significance of these numbers as follows: more Maths DNA Mitochondrial Base Pairs In Chamakam Rudram

    The Vedas are not mathematical texts; they are merely hymns to the Vedic gods. However, the word Veda means “knowledge”, and when analyzed closely it actually contains many mathematical references, especially in the section on Jyotisa, or “the constellations”. Unfortunately, almost all of these references are implied, so much of the interpretation is largely guesswork. Another reason that the Vedas are hard to interpret is that because it was an oral document, there are no symbols for numbers or operations — only words. It is highly likely, however, that they did use symbols, because without them math becomes very tedious. For example, consider doing a multiplication problem using “four thousand six hundred and thirty-seven times two hundred and eighty-eight.” You would most likely convert the words to symbols, do the math on a piece of paper, and then probably only take the time to convert the answer back into words…..And yet another oddity of the Sanskrit language involves what happens with compound numbers, numbers with more than one “digit” (like “thirty-four”). In normal Sanskrit, compound words (like “servant of the king”) came from left to right in order of prevalence (so our example would be “king-servant”; “servant-king” would mean a servant who was treated well). However, compound numbers are written the opposite way, with the higher digits on the right. (Our number 529 would be written “nine-two-five”.) It is always like this, and there is even a rule included: ankanam vamato gatih, which literally means “the understanding of the numbers in the reverse way…

    A good example of a story from which we can extract mathematics is one about a man named Manu.3 Manu had ten wives, who had one, two, three, four, etc. sons each (the first wife had one son, the second wife two, etc.). The one son allied with the nine sons, and the two sons allied with the eight, and so on until the five sons were left by themselves. They asked Manu for help, and so he gave them each a samidh or “oblation-stick”. The five sons then used these sticks to defeat all of the other sons.

    On the surface, this is just a silly fable, but it shows several things about Vedic mathematics. Because the ten sons did not ally with anyone, and the nine did with the one, eight with two, and so on, the mathematicians must have been thinking that nine plus one, and eight plus two equal ten. This obviously shows that they practiced addition, and it also implies that they used a base 10, or decimal, system. For the second part of the story, the authors probably added the tens up to find that there were 50 allied sons. When the five remaining sons asked their father for help, it is likely that he gave them just enough mathematical power to defeat the others. This would mean that each stick equaled the strength of 10 men, for a total of 50. With the five sons added to that, they were able to defeat the 50. (Or maybe the father gave them 50 men worth of sticks, thinking it would be an equal battle, but not realizing that the five sons would throw off the balance.) But this 50 business implies both multiplication and division as well, because there were five groups of ten sons allied, or five times ten. Then, when the father went to decide the power of the sticks, he would have had to divide that 50 by five, to split the power equally among the five sons…even further, the story can be shown to symbolize the idea of positional notation — the idea of place values in numerals. (For an example of positional notation, 218 is the same as 200 + 10 + 8, or 2 x 102 plus 1 x 101 plus 8 x 100. In summary, the order of numerals tells how big the numbers are.) The “oblation-sticks” are obviously thought of as very powerful, just as 10 might be thought of as more “powerful” than a lowly 1. So when the 5 “lowly” sons were “added” to the 5 “powerful” sticks, this could have symbolized the 50 and 5 making 55, which is a bigger number (and therefore more powerful) than 50. This view of things also gives further evidence to the fact that they used base 10.

    So this simple story shows examples of addition, multiplication, division, base 10, and even positional notation. The Vedas are full of these stories, and many more examples are given throughout of all these concepts, along with subtraction, fractions, and squares.5 There are even instances of arithmetic and geometric sequences, which are series of numbers that increase by adding or multiplying a certain number (arithmetic sequence: 2 (+3 =) 5, 8, 11, 14, … ; geometric sequence: 2 ( x 3 =) 6, 18, 54, … Source. Zimmerman, Francis. “Lilavati, Gracious Lady of Arithmetic.” UNESCO Courier. Nov, 1989, p 20. [bibliography entry]
    2 Pandit, M. D. Mathematics as Known to the Vedic Samhitas. Dehli, India: Sri Satguru Publications, 1993, p 153. [bibliography entry]

    3 From Maitrayani Samhita (part of the Vedas), section 1.5.8, trans. M. D. Pandit. [bibliography entry]

    4 Most of the ideas in this paragraph come from Pandit, p 102.

    5 See Pandit for more details

    Going

    A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The most commonly used system of numerals is decimal. Indian mathematicians are credited with developing the integer version, the Hindu–Arabic numeral system.The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the number seven would be represented by ///////….The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as +++ //// and the number 123 as + − − /// without any need for zero. This is called sign-value notation. More elegant is a positional system, also known as place-value notation. Again working in base 10, ten different digits 0, …, 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1 or more precisely 3×102 + 0×101 + 4×100. Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to “skip” a power. The Hindu–Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.In computers, the main numeral systems are based on the positional system in base 2 (binary numeral system), with two binary digits, 0 and 1. Source.https://en.m.wikipedia.org/wiki/Numeral_system
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  • Mathematical Concepts In Vedas Sulaba Sutras

    Mathematical Concepts In Vedas Sulaba Sutras

    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!
    Vedic Mathematics in Sulaba Sutras.
    Sulaba Sutras of Kathyayana.

    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, 972x2 = 972 + m for example, found in one of the Samhitas. Sites Google

    Sulba Sutras

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

    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

    1. 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].”
    2. ^ a b c Plofker (2007), p. 387
    3. ^ a b Pingree (1981), p. 4
    4. ^ a b Plofker (2009), p.18
    5. ^ Plofker (2009), p. 11
    6. ^ Pingree (1981), p. 6
    7. ^ Delire (2009), p. 50
    8. ^ Staal (1999), p. 111
    9. ^ Plofker (2009), p 19.
    10. ^ Bürk (1901), p. 554
    11. ^ Heath (1925), p. 362
    12. ^ “Square Roots of Sulbha Sutras”. pi.math.cornell.edu. Retrieved 2020-05-24.
    13. ^ 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.
    14. ^ Gupta (1997), p. 154
    15. ^ Staal (1999), pp. 106, 109–110
    16. ^ Seidenberg (1978)
    17. ^ van der Waerden (1983)
    18. ^ Van der Waerden, Barten L (1983). Geometry and Algebra in Ancient Civilizations. Springer Verlag. p. 12. ISBN 0387121595.
    19. ^ 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.
    20. ^ 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.”
    21. ^ Krishnan, K S (2019). Origin of Vedas, Chapter 5. Notion Press. ISBN 978-1645879800.
    22. ^ Seidenberg (1983), p. 121
    23. ^ Pingree (1981), p. 5
    24. ^ Plofker (2009), p. 17
    25. ^ Thibaut (1875), pp. 232–238
    26. ^ Plofker (2007), pp. 388–389
    27. ^ Boyer (1991), p. 207
    28. ^ Joseph, G.G. (2000). The Crest of the Peacock: The Non-European Roots of Mathematics. Princeton University Press. p. 229. ISBN 
    29. ^ Thibaut (1875), pp. 243–246
    30. ^ a b Plofker (2007), pp. 388-391
    31. ^ a b c Plofker (2007), p. 391
    32. ^ Plofker (2007), p. 392, 
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  • Infinity And Nothing Indian Mathematics

    Infinity And Nothing Indian Mathematics

    This is my fourth article in the series on History of Mathematics.  If the approach of Vedas to knowledge is different from the western concept of knowledge ( I have explained this in my last article), the approach for gaining such knowledge ,Apara , relating to mundane lifetime activities, which include Mathematics,in Indian system’ basic approach to basic axiom of Knowledge is totally different from the western concept.

    Western system of thought has the basic concept as ‘ Ex Nihilo Nihil Fit’ – Out of Nothing, Nothing Comes’.

    But Indian system’s axiom is 

    This is full,That is Full,Having Taken Full out of Full, Full remains Full.
    ॐ पूर्णमदः पूर्णमिदं पूर्णात्पूर्णमुदच्यते ।
    पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते ॥
    ॐ शान्तिः शान्तिः शान्तिः ॥
    Om Puurnnam-Adah Puurnnam-Idam Puurnnaat-Puurnnam-Udacyate |
    Puurnnasya Puurnnam-Aadaaya Puurnnam-Eva-Avashissyate ||
    Om Shaantih Shaantih Shaantih ||

    Meaning:
    1: Om, That (Outer World) is Purna (Full with Divine Consciousness); This (Inner World) is also Purna (Full with Divine Consciousness); From Purna is manifested Purna (From the Fullness of Divine Consciousness the World is manifested),
    2: Taking Purna from Purna, Purna indeed remains (Because Divine Consciousness is Non-Dual and Infinite),
    3: Om, Peace, Peace, Peace. Source.Green Message Isa Upanishad, Yajurveda (400 BC)

    What is applicable to Metaphysics is applicable to Physics in Indian Thought.So, we have totally varied approaches to knowledge,but the essence is the same, except that in the western axiom it is Nihilism and in Indian Thought ,it is Positivism.

    Hence difference in paths traveled by Indian mathematics and other sciences is different from the West.

    In Indian Thought, Darkness is a positive concept,absence of darkness is Light and not the other way; Acquisition of something from outside sources is Knowledge in the West and is a positive concept.

    We find two Concepts in Vedic Thought. That of Infinity and Sunya. Infinity is limitless, beyond description and comprehension for it is beyond our mind which is shackled by Space and Time.Indian thought considers Infinity as a positive concept and ‘Finite’ is a negative concept in that What is Infinite is Real and what is Finite is because of our Limitations in Perception. On Infinity and what is Finite is a topic that needs a detailed analysis and it would be out of place here.

    Now onto Nothingness.

    Nadadiya Suktha is the 129th hymn of the 10th mandala of the Rigveda (10:129). It is concerned with cosmology and the origin of the universe. Source.Nasadiya Sukta

    Nasadiya Sukta (Hymn of non-Eternity, origin of universe):

    There was neither non-existence nor existence then;
    Neither the realm of space, nor the sky which is beyond;
    What stirred? Where? In whose protection?

    There was neither death nor immortality then;
    No distinguishing sign of night nor of day;
    That One breathed, windless, by its own impulse;
    Other than that there was nothing beyond.

    Darkness there was at first, by darkness hidden;
    Without distinctive marks, this all was water;
    That which, becoming, by the void was covered;
    That One by force of heat came into being;

    Who really knows? Who will here proclaim it?
    Whence was it produced? Whence is this creation?
    Gods came afterwards, with the creation of this universe.
    Who then knows whence it has arisen?

    Whether God's will created it, or whether He was mute;
    Perhaps it formed itself, or perhaps it did not;
    Only He who is its overseer in highest heaven knows,
    Only He knows, or perhaps He does not know.

    —Rigveda 10.129 (Abridged, Tr: Kramer / Christian source.

    Nothingness – Is it a positive entity? Absence og Attributes is an Attribute.

    And Infinity has directions.

    Surya Prajnapti which is thought to be around the 4th century BC and the Jambudvipa Prajnapti from around the same period, have recently received attention through the study of later commentaries. The Bhagabati Sutra dates from around 300 BC and contains interesting information on combinations. From about the second century BC is the Sthananga Sutra which is particularly interesting in that it lists the topics which made up the mathematics studied at the time. In fact this list of topics sets the scene for the areas of study for a long time to come in the Indian subcontinent. The topics are listed in [2] as:-

    … the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.

    The ideas of the mathematical infinite in Jaina mathematics is very interesting indeed and they evolve largely due to the Jaina’s cosmological ideas. In Jaina cosmology time is thought of as eternal and without form. The world is infinite, it was never created and has always existed. Space pervades everything and is without form. All the objects of the universe exist in space which is divided into the space of the universe and the space of the non-universe. There is a central region of the universe in which all living beings, including men, animals, gods and devils, live. Above this central region is the upper world which is itself divided into two parts. Below the central region is the lower world which is divided into seven tiers. This led to the work described in on a mathematical topic in the Jaina work, Tiloyapannatti by Yativrsabha. A circle is divided by parallel lines into regions of prescribed widths. The lengths of the boundary chords and the areas of the regions are given, based on stated rules.

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    The Jaina cosmology contained a time period of 2^{588}2588 years. Note that 2^{588}2588 is a very large number!

    2^{588}2588 = 1013 065324 433836 171511 818326 096474 890383 898005 918563 696288 002277 756507 034036 354527 929615 978746 851512 277392 062160 962106 733983 191180 520452 956027 069051 297354 415786 421338 721071 661056…….
    Infinity Jainism

    Jaina mathematics recognised five different types of infinity [2]:-

    … infinite in one direction, infinite in two directions, infinite in area, infinite everywhere and perpetually infinite.

    The Anuyoga Dwara Sutra contains other remarkable numerical speculations by the Jainas.

    For example several times in the work the number of human beings that ever existed is given as 2^{96}296. Source. Jaina Mathematics ‘ Accessibility Statement.’https://mathshistory.st-andrews.ac.uk/Miscellaneous/accessibility/

    These Jain texts date from 4th Century BC.

    Jainism is a part of Indian philosophical system ,though it does not recognise Vedas as a source of Knowledge,much like Buddhism. These are called Nastika systems of Indian philosophy along with Carvaka,Ajivaka,Mimamsa, Vaiseshika and Nyaya systems.

    More articles on Vedic Mathematical Concepts follow.
  • Vedic Sutras 29 On Mathematics Explained.With Translator

    Vedic Sutras 29 On Mathematics Explained.With Translator

    This is the third part of my series on History of Mathematics. In my earlier articles, I had explored the concept of mathematics and how and where it emerged from. Though mathematics had made its way in India very early ,in Vedic texts,among mathematicians who lived before Christ,in Jainism/ Buddhism and in the language I am familiar with (in which I can read ancient texts)Tamil. I shall write about them. Now I shall attempt at the references to Mathematics in Vedic texts. One may note that the oldest literature known to man, Rig Veda is dated about 5000 years. My view is that it is older at least by few thousand years. I had written articles on this topic in this blog.

    Vedas are called the Sabda Pramana. It means that Vedas are one of the tools of knowledge and it represents testimony. Vedas are Testimonies of Seekers of Truth and they had recorded their experiences in the Form of Sound. Vedas are one of the tools of knowledge recognised by Hinduism,along with Perception, Inference,Comparison ,Logic,Intuition,and Experience.

    Sanatana Dharma recognised two levels of Knowledge.One is Para Vidya which is Absolute and another is Apara .It is lower knowledge in that it deals knowledge that is applicable and useful for day to day activities. It is lower knowledge as any thing that binds one to the world in the form of Birth ,death and the cycle involved , is treated as Inferior Knowledge.The knowledge that helps one realise Self is Real Knowledge and it is Absolute.

    So references to Sciences as we know of today , as defined by Western thought, are not pursued as distinctly as the western system of Education. The information relating to these concepts are made in the passing so long they are relevant in Realising Brahman .So the scientist concepts as we know of know today are found in as much as they are found useful in the conduct of Vedic Rituals ,when King’ Prowess or skills are explained in Purana or Ithihasa. And in many an instance, the principles are inferred either from the results or understanding the Aphorisms. And this needs special skill set. As Nature is personified Attributes are ascribed and they reveal the level of scientific knowledge of the Vedic people. For instance, Sun is called Surya and Attributes of Surya include the composition of Light, movement of Sun in the Universe.One can find information about the planets, Galaxies,Stars , Blackholes, Wormhole, Concept of Time as Cyclic, which is now validated by Quantum Theory.And we have the Vaiseshika system deals extensively about Atomic Theory, Particles,sub particles. Nyaya system deals with advanced Logical systems. There are concepts which are applicable to Computer Languages. Binary principles, Binomial theorem, Pythogoras Theorem Trigonometry,Calculus…all are found in ancient Indian texts.

    You may have noticed that I have referred to some systems which are not a part of Vedas. Reason is that Vedas alone do not contribute to Hinduism. Vedas are a Source of Knowledge. Other systems like Nyaya,Vaiseshika Sankhya,Yoga,Purvamimasa and Uttaramimasa are other systems of Indian philosophy. Of these Uttara Mimamsa is Upanishads which is a part of Vedas.Having this point in our approach, let us see what Vedas mention ,in the passing, about scientific, especially Mathematical Concepts.

    Chamaka expresses Mitochondrial Pairs of DNA on the form of Mathematics.

    Whe have, Vedic Mathematics. This helps one to handle fundamental mathematics easily.

    Vedic Mathematics Basic Sixteen Sutras, Aphorisms.
    1. Ekadhikina Purvena
      (Corollary: Anurupyena)
      Meaning: By one more than the previous one
    2. Nikhilam Navatashcaramam Dashatah
      (Corollary: Sisyate Sesasamjnah)
      Meaning: All from 9 and the last from 10
    3. Urdhva-Tiryagbyham
      (Corollary: Adyamadyenantyamantyena)
      Meaning: Vertically and crosswise
    4. Paraavartya Yojayet
      (Corollary: Kevalaih Saptakam Gunyat)
      Meaning: Transpose and adjust
    5. Shunyam Saamyasamuccaye
      (Corollary: Vestanam)
      Meaning: When the sum is the same that sum is zero
    6. (Anurupye) Shunyamanyat
      (Corollary: Yavadunam Tavadunam)
      Meaning: If one is in ratio, the other is zero
    7. Sankalana-vyavakalanabhyam
      (Corollary: Yavadunam Tavadunikritya Varga Yojayet)
      Meaning: By addition and by subtraction.Puranapuranabyham
    8. Puranapuranabyham(Corollary: Antyayordashake’pi Meaning: By the completion or non-completion.
    9. Chalana-Kalanabyha(Corollary: Antyayoreva)Meaning: Differences and Similarities.
    10. Yaavadunam(Corollary: Samuccayagunitah)Meaning: Whatever the extent of its deficiency.
    11. Vyashtisamanstih(Corollary: Lopanasthapanabhyam)Meaning: Part and Whole.
    12. Shesanyankena Charamena(Corollary: Vilokanam)Meaning: The remainders by the last digit.
    13. Sopaantyadvayamantyam(Corollary: Gunitasamuccayah Samuccayagunitah)Meaning: The ultimate and twice the penultimate.
    14. Ekanyunena Purvena(Corollary: Dhvajanka)Meaning: By one less than the previous one.
    15. Gunitasamuchyah(Corollary: Dwandwa Yoga)Meaning: The product of the sum is equal to the sum of the product.
    16. Gunakasamuchyah(Corollary: Adyam Antyam Madhyam Meaning: The factors of the sum is equal to the sum of the factors. Source.Das, Subhamoy. “The 16 Sutras of Vedic Math.” Learn Religions, Aug. 26, 2020, learnreligions.com/vedic-math-formulas-177068 0. https://www.learnreligions.com/vedic-math-formulas-1770680

    Check these sites.

    Vedic Maths in Hindi
    Vedic Mathematics English

    The secularists dismiss this as Tricks!

    These statements have been since rejected in their entirety. Krishna Tirtha failed to produce the sources, and scholars unanimously note it to be a mere compendium of tricks for increasing the speed of elementary mathematical calculations with no overlap with historical mathematical developments during the Vedic period. However, there has been a proliferation of publications in this area and multiple attempts to integrate the subject into mainstream education by right-wingHindu nationalist governments. https://en.m.wikipedia.org/wiki/Vedic_Mathematics

    Does Truth Matter? I Offer A ₹5,00,000 INR ‘Vedic’ Maths Reward  … For Serious Primary Evidence And Proof The 16 Mathematics Sutras Of Bharati Krishna Tirtha Irrefutably Exist In Any Extant Written Vedas.…

    Jagadguru Shankaracharya Swami Bharati Krishna Tirtha’s view was that Vedic maths is NOT to be approached from a factual standpoint. If it is accepted that the Vedas are the source of all knowledge, then the 16 Sutras of Vedic Maths should have been contained in them.

    Dr. V.S. Agrawala, friend of Bharati Krishna Tirtha and Editor of the book titled ‘Vedic Maths’.
    Are The 16 Sutras Of Vedic Maths From India’s Ancient Vedas? What Do Indian Scholars Say?
    VEDIC SCHOLAR PROF. DR. SUBHASH KAK

    “First question: What is Vedic mathematics? It is not mathematics from the Vedic period. Rather it consists of many clever mathematical sutras and algorithms that were devised by Swami Bharati Krishna Tirtha (1884–1960) who for a long time was the Shankaracharya of Govardhan Matha in Puri.

    The teaching of Vedic Mathematics cannot be justified on the grounds that it tells the students something about India’s ancient mathematical heritage. This heritage without even counting the invention of zero was brilliant and unique but Swami Bharati Krishna Tirtha’s Vedic mathematics has nothing to do with it.” Source…

    MATHEMATICIAN PROF. DR. C. K. RAJU

    “Advocating ‘Vedic mathematics’ as a replacement for traditional Indian arithmetic is hardly an act of nationalism; it only shows ignorance of the history of mathematics.

    But where in the Vedas is “Vedic mathematics” to be found? Nowhere. Vedic mathematics has no relation whatsoever to the Vedas. It actually originates from a book misleadingly titled Vedic Mathematics by Bharati Krishna Tirtha. The book admits on its first page that its title is misleading and that the (elementary arithmetic) algorithms expounded in the book have nothing to do with the Vedas. This is repeated on p. xxxv: “Obviously these formulas are not to be found in the present recensions of Atharvaveda.”. https://kreately.in/is-vedic-maths-vedic/

    And more from ‘open’ site.

    As stories go, this is not a bad one, but the evidence does nothing to support it. The 16 sutras expounded by Tirathji do not appear in any known edition of the Atharva Veda. Tirathji’s defenders have claimed that Tirathji was so immersed in Vedic thought that he was able to glean what the Vedic seers had in mind even if it was not explicitly so stated anywhere in the Vedic corpus. If one were to actually concede this meeting of minds between Tirathji and the ancient Vedic seers, it would have the unfortunate consequence of implying that not just Tirathji but even these seers were limited in their mathematical understanding.

    All the sutras largely do is make the burden of addition and multiplication faster (though never nearly as fast as the cheapest pocket calculator), and even that, they do at a cost. Students studying the traditional method of multiplication should ideally understand (and bad teachers themselves fail to grasp this) what multiplication is, how it works, and how it is in essence an act of repeated addition. Tirathji’s methods are just rules that make mathematics seem like a bunch of tricks which are easy to implement but difficult to understand.

    Take, for example, the multiplication of 9 and 7. Line them along with their difference from 10. That is:

    9–10 = –1 and 7–10 = –3

    9–1

    7–3

    ——

    6   3

    You obtain the answer in the following fashion: the unit’s digit is the two differences multiplied together, –1 x –3 = 3 and the other digit 6 is just either of the diagonals added together, that is, 9–3 = 7–1 = 6. This method can be extended to much larger numbers. It is a neat trick, but it does not make multiplication easier to fathom, quite the contrary….open the magazine

    None of these scholars have disproved the fact that Vedic Mathematics sutras work.

    How ridiculous can one be? The author of the book produced Non sense? The formulas work and as I mentioned at the beginning of this article that to understand Indian science, one has to understand Hinduism ‘s Concept of Knowledge first, before jumping into right wing bashing.

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  • Mathematicians of India Before Christ

    Mathematicians of India Before Christ

    This is in continuation of my earlier article History of Mathematics Preface Bakshali Manuscript.We may now look some ancient Indian treatises on Mathematics.The names that come to one’ s mind are Aryabhatta and Varahamihira.

    Mathematics, Numerals in Brahmi text, India.
    Mathematics in India

    Aryabhatta

    Aryabhata (Sanskrit: आर्यभट, ISO: Āryabhaṭa) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the Āryabhaṭīya (which mentions that in 3600 Kali Yuga, 499 CE, he was 23 years old) and the Arya-siddhanta.Aryabhata (Sanskrit: आर्यभट, ISO: Āryabhaṭa) or Aryabhata I(476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the Āryabhaṭīya (which mentions that in 3600 Kali Yuga, 499 CE, he was 23 years old) and the Arya-siddhanta.…

    His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers , algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.

    The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata’s contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara.

    Brahmagupta

    Brahmagupta (c. 598 – c. 668 CE) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta (BSS, “correctly established doctrine of Brahma”, dated 628), a theoretical treatise, and the Khaṇḍakhādyaka (“edible bite”, dated 665), a more practical text.

    • Brāhmasphuṭasiddhānta,composed in 628 CE.
    • Khaṇḍakhādyaka,composed in 665 CE.
    • Grahaṇārkajñāna,(ascribed in one manuscript.

    Lalla

    Lalla (c. 720–790 CE) was an Indian mathematician, astronomer, and astrologer who belonged to a family of astronomers.

    Śiṣyadhīvṛddhidatantra is his work.

    Panini 5th Century BC

    We know Panini to be the grammarian of Sanskrit Language . He was an illustrious Mathematician too. His notation was similar to modern mathematical notation, and used metarules, transformations, and recursion.Pingala (roughly 3rd–1st centuries BC) in his treatise of prosody uses a device corresponding to a binary numeral system. His discussion of the combinatorics of meters corresponds to an elementary version of the binomial theorem. Pingala’s work also contains the basic ideas of Fibonacci numbers. Read this Mathematical Structures of Ashtadyayi

    Pingala 3- 2 Century BC

    Another great mathematician of ancient India.ancient Indian poet and mathematician, the author of the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Metres.( Chandas,Vedic Metres).The Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables. The discussion of the combinatorics of meter corresponds to the binomial theorem. Halāyudha’s commentary includes a presentation of Pascal’s triangle (called meruprastāra). Pingala’s work also includes material related to the Fibonacci numbers, called mātrāmeru.

    The Surya Siddhanta,

    Homage to Brahma, Suryasiddhantha

    The Surya Siddhanta ‘The text is known from a 15th-century CE palm-leaf manuscript, and several newer manuscripts. It was composed or revised c. 800 CE from an earlier text also called the Surya Siddhanta.Now date is arbitrarily assigned as 4/5 Century BC!(According to al-Biruni, the 11th-century Persian scholar and polymath, a text named the Surya Siddhanta was written by one Lāta.The second verse of the first chapter of the Surya Siddhanta attributes the words to an emissary of the solar deity of Hindu mythology, Surya, as recounted to an asura (a mythical being) called Maya at the end of Satya Yuga, the first golden age of Hindu mythology, around two million years ago.Maya has Tamil connection. I shall detail this in another post.The Surya Siddhanta thus consists of cryptic rules in Sanskrit verse. It is a compendium of astronomy that is easier to remember, transmit and use as reference or aid for the experienced, but does not aim to offer commentary, explanation or proof. The text has 14 chapters and 500 shlokas. It is one of the eighteen astronomical siddhanta (treatises), but thirteen of the eighteen are believed to be lost to history. The Surya Siddhanta text has survived since the ancient times, has been the best known and the most referred astronomical text in the Indian tradition. Suryasiddhantha

    References. https://en.m.wikipedia.org/wiki/History_of_mathematics

    Shall write on Vedic Mathematical Structures, to be followed by Tamil concepts on Mathematics.

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