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Indian contributions to mathematics Indian contributions to mathematics Presentation Transcript

  • Indian contributions to mathematics  7 1 5 1 3 1 1 4  - Dr. Bhaskar Kamble
  • Generally accepted world view Greece European Renaissance Modern Science  “Most of the amazing science and technology knowledge systems of the modern world are credited to have started around the time of the Renaissance movement in Europe around the 15th century. These knowledge systems are generally traced back to roots in the civilization of Ancient Greece, and occasionally, that of Ancient Egypt. Hence, most of the heroes we are taught about in school and college are European, or Greek.”* *http://bharathgyanblog.wordpress.com/2013/09/21/calculus-was-discovered-in-india/
  • A brief history of science generally accepted today View slide
  • Humans originate from the apes A brief history of science generally accepted today View slide
  • Humans originate from the apes ancient civilizations (Sumerian, Egyptian, Babylonian etc.) A brief history of science generally accepted today
  • ancient civilizations (Sumerian, Egyptian, Babylonian etc.) Around 500 B.C. Origin of mathematics and philosophy in ancient Greece Precursor to the European Renaissance MAJOR STEP I Humans originate from the apes A brief history of science generally accepted today
  • ancient civilizations (Sumerian, Egyptian, Babylonian etc.) Around 500 B.C. Origin of mathematics and philosophy in ancient Greece. Precursor to the European Renaissance Dark ages in Europe MAJOR STEP I Humans originate from the apes A brief history of science generally accepted today
  • Renaissance in Europe, based on the Greek civilization ancient civilizations (Sumerian, Egyptian, Babylonian etc.) Around 500 B.C. Origin of mathematics and philosophy in ancient Greece. Precursor to the European Renaissance MAJOR STEP I MAJOR STEP II Dark ages in Europe Humans originate from the apes A brief history of science generally accepted today
  • Renaissance in Europe, based on the Greek civilization ancient civilizations (Sumerian, Egyptian, Babylonian etc.) Around 500 B.C. Origin of mathematics and philosophy in ancient Greece. Precursor to the European Renaissance The birth of the modern sciences and mathematics- Kepler, Galileo, Newton etc. are the heroes. MAJOR STEP I MAJOR STEP II Dark ages in Europe A brief history of science generally accepted today Humans originate from the apes
  • MAJOR STEP I MAJOR STEP II Retaining the skeletal structure… The birth of the modern sciences and mathematics- Kepler, Galileo, Newton etc. are the heroes. Renaissance in Europe, based on the Greek civilization Dark ages in Europe Around 500 B.C. Origin of mathematics and philosophy in ancient Greece. Precursor to the European Renaissance ancient civilizations (Sumerian, Egyptian, Babylonian etc.) Humans originate from the apes A brief history of science generally accepted today
  • Renaissance in Europe, based on the Greek civilization The birth of the modern sciences and mathematics- Kepler, Galileo, Newton etc. are the heroes. 500 B.C. Mathematics and philosophy in Greece. A brief history of science generally accepted today
  • Renaissance in Europe, based on the Greek civilization The birth of the modern sciences and mathematics- Kepler, Galileo, Newton etc. are the heroes. 500 B.C. Mathematics and philosophy in Greece. WHAT IS BEING GENERALLY MISSED IN THE ABOVE … A brief history of science generally accepted today
  • Renaissance in Europe, based on the Greek civilization The birth of the modern sciences and mathematics- Kepler, Galileo, Newton etc. are the heroes. 500 B.C. Mathematics and philosophy in Greece.
  • Renaissance in Europe, based on the Greek civilization Surya Siddhanta, (2000 B.C. or earlier) Sulbasutras, 800 B.C. or earlier Astronomical text with trigonometric, algebraic and geometric principles Geometry and algebra for making Vedic altars, statement and demonstration of the Pythagoras theorem 300 years before Pythagoras. Aryabhatta 500 B.C. Mathematics and philosophy in Greece. Brahmagupta, Bhaskara I Bhaskara II Kerala school of mathematics, (Madhava, Nilakantha, Jyeshthadeva etc.) – used differential and integral calculus 200 years before Newton and Leibniz, infinite series, spherical trigonometry, astronomy etc. The birth of the modern sciences and mathematics- Kepler, Galileo, Newton etc. are the heroes.
  • Renaissance in Europe, based on the Greek civilization Surya Siddhanta, (2000 B.C. or earlier) Sulbasutras, 800 B.C. or earlier Astronomical text with trigonometric, algebraic and geometric principles Geometry and algebra for making Vedic altars, statement and demonstration of the Pythagoras theorem 300 years before Pythagoras. Aryabhatta 500 B.C. Mathematics and philosophy in Greece. Brahmagupta, Bhaskara I Bhaskara II Kerala school of mathematics, (Madhava, Nilakantha, Jyeshthadeva etc.) – used differential and integral calculus 200 years before Newton and Leibniz, infinite series, spherical trigonometry, astronomy etc. The birth of the modern sciences and mathematics- Kepler, Galileo, Newton etc. are the heroes. Also most of the high school mathematics attributed to the Greeks. Algebra, geometry, trigonometry, the Pythagoras theorem, Trigonometry, decimal system, concept of zero, rational, irrational and negative numbers, algebraic identities comes from India!
  • “As for India, it would appear that it has played a minimal role in this magical story. Hence, many western accounts of the “Ascent of Man” do not devote even a single line to India’s contributions.”* *http://bharathgyanblog.wordpress.com/2013/09/21/calculus-was-discovered-in-india/
  • Renaissance in Europe, based on the Greek civilization Surya Siddhanta, (2000 B.C. or earlier) Sulbasutras, 800 B.C. or earlier Astronomical text with trigonometric, algebraic and geometric principles Geometry and algebra for making Vedic altars, statement and demonstration of the Pythagoras theorem 300 years before Pythagoras. Aryabhatta 500 B.C. Mathematics and philosophy in Greece. Brahmagupta, Bhaskara I Bhaskara II Kerala school of mathematics, (Madhava, Nilakantha, Jyeshthadeva etc.) – used differential and integral calculus 200 years before Newton and Leibniz, infinite series, spherical trigonometry, astronomy etc. The birth of the modern sciences and mathematics- Kepler, Galileo, Newton etc. are the heroes. Also most of the high school mathematics attributed to the Greeks. Algebra, geometry, trigonometry, the Pythagoras theorem, Trigonometry, decimal system, concept of zero, rational, irrational and negative numbers, algebraic identities comes from India!
  • Some of the references used
  • Some of the references used (Referred to as “Encyclopaedia” in the presenation)
  • (referred to as “TAGS” in the presentation)
  • India’s contributions to mathematics Part 1
  • THE CASE OF THE “PYTHAGORAS THEOREM”
  • From an article published in The Transactions of the Royal Society of Edinburgh, 1790 by John Playfair “Remarks on the Astronomy of the Brahmins” THE CASE OF THE “PYTHAGORAS THEOREM”
  • This article discusses: Three sets of astronomical tables obtained from India dated to the end of the Kali Yuga (3102 BC.) THE CASE OF THE “PYTHAGORAS THEOREM” (1) Brought from Siam in 1687 to Europe (2) Sent from Krishnapuram in about 1750 to Europe (3) Brought from Tiruvallur in 1772 from Europe
  • From ‘Hints concerning the observatory at Benares’ c. 1783 by Reuben Burrow “we know that he (Pythagoras) went to India to be instructed; but the capacity of the learner determines his degree of proficiency, and if Pythagoras on his return had so little knowledge in geometry as to consider the forty-seventh of Euclid as a great discovery, he certainly was entirely incapable of acquiring the Indian method of calculation, through his deficiency of preparatory knowledge … …each teacher, or head of sect that drew his knowledge from Indian sources, might conceal his instructors to be reckoned an inventor.” THE CASE OF THE “PYTHAGORAS THEOREM”
  • Sulba Sutras Baudhayana (from the Krishna Yajurveda) Apastamba (from the Krishna Yajurveda) Katyayana (from the Shukla Yajurveda) Composed around 800 BC. (although the authors emphasize they are merely stating facts known since the early Vedic age.) THE CASE OF THE “PYTHAGORAS THEOREM” “Pythagoras theorem” in the Vedas
  • - Dr. T. A. Sarasvati Amma, “Geometry in ancient and medieval India” (Motilal Banarsidass, Delhi (2007) ) THE CASE OF THE “PYTHAGORAS THEOREM”
  • - Dr. T. A. Sarasvati Amma, “Geometry in ancient and medieval India” (Motilal Banarsidass, Delhi (2007) ) THE CASE OF THE “PYTHAGORAS THEOREM”
  • Around 2,500 years ago Pythagoras went from Samos to the Ganga to learn Geometry (evidence and reference will be shown later). He would certainly not have undertaken such a strange journey had the reputation of the Brahmin’s science not been long established in Europe. Pythagoras and his followers were vegetarians and believed in transmigration of souls, both of which are Indian concepts (many more similarities of philosophical type will be shown later). THE CASE OF THE “PYTHAGORAS THEOREM”
  • Pythagoras theorem should be called Baudhayana theorem! (at least in India)
  • Trigonometry in the Surya Siddhanta An astronomical text dated to 2000 BC: some extracts from “On the trigonometric tables of the Brahmins” by John Playfair published in the Transactions of the Royal Society of Edinburgh, Vol. IV, 1798
  • This article discusses a table of sines calculated for different angles in the Surya Siddhanta and the possible working principle behind it
  • Working principle for constructing the table of sines in the Surya Siddhanta, as deduced by Playfair  sin)sin()cos2()2sin(  1. Suppose sinθ, sin(θ+α), and cos α are known. Then from the above equation we can find sin(θ+2α). 2. Next in the above equation replace θ by θ+α. So we get 3. sin(θ+2α) is known from the previous equation. So from the above we get sin(θ+3α). 4. Again following the same procedure, we get sin(θ+4α), and so on…. )sin()2sin()cos2()3sin(  
  • BABABA cossin2)sin()sin(  Proof   2  BA BA     B A  cos)sin(2)2sin(sin   sin)sin()cos2()2sin(  
  •  sin)sin()cos2()2sin( 
  • Regarding the antiquity of the trigonometry contained in the Surya Siddhanta
  • Pingala, Meru Prastara and the Binomial theorem “Math for poets and drummers”, R. W. Hall, Dept. of Mathematics and Computer Science, St. Joseph’s University, Philadelphia “Binomial theorem in ancient India” A. K. Bag, Indian Journal of History of Science 1966  Pinagala, a scholar studying the mathematics of music and rhythm, described the Meru Prastara in his treatise Chhandah-Sutra in 200 B.C. (chhandah= meter of a poem, e.g. Bhagavad Gita chapter 10.35).  In the process he described the binomial theorem for integer index in 200 B.C. several centuries earlier than anywhere else in the world.  It is also described by Halayudha’s commentary on the Chandah sutras dating to the 10th century AD.  Today it is known as Pascal’s triangle after the posthumous publication of Traité du triangle arithmétique in 1665.
  • Other contributions by scholars of music, language, and rhythm • The sequence of numbers 0,1,1,2,3,5,… were first given by Virahanka (ca. 600-800 A.D.), Gopala (earlier than 1135 AD) and Acharya Hemachandra (1150 AD). about 50 years before Fibonacci. (Today they are called Fibonacci numbers). (“The so-called Fibonacci numbers in ancient and medieval India”, P. Singh, Historia Mathematica 12, 229-244 (1985).) • In computer science, the notation technique known as Backus-Naur form was first described by Panini, (a linguist and Sanskrit grammarian from 4th century BC born in Pushkalavati, Gandhara, (now in Pakistan)). The works of modern day linguists and information theorists such as Leonard Bloomfield, Zellig Harris, Axel Thue, Emil Post, Alan Turing, Noam Chomsky, and John Backus, are based extensively on Panini’s works. • Panini also anticipated the binary number system. “On some rules of Panini”, Leonard Bloomfield, Journal of the American Oriental Society, 47, 61 (1927). “ ‘Panini-Backus’ form suggested ”, P. Z. Ingerman, Comm. of the ACM, 1967.
  • General Observations  Techniques of the fundamental arithmetic operations: addition, subtraction, multiplication, division; Extracting square and cube roots; the rules of operations with fractions and surds;  the Indian methods of performing long multiplications and divisions were introduced in Europe as late as the 14th century AD…  The rule of three, brought to Europe via the Arabs (from India) came to be known as the Golden rule for its great popularity and utility in commercial computations…  Modern methods of extracting square and cube roots, described by Aryabhata in the 5th century AD, were introduced in Europe only in the 16th century AD.  The introduction of negative numbers and systematic use of symbols to denote unknown quantities and arithmetic operations … the development of the algebraic formalism.
  • And of course… • (wrongly called Arabic numerals – should be called the Indian number system) Whole numbers, rational numbers, irrational numbers to any degree of accuracy, addition, subtraction, multiplication, division, square roots, cube roots, can express incredibly small or incredibly large numbers. (For e.g. 888 in Roman system is DCCCLXXXVIII ). • “It is no coincidence that the mathematical and scientific renaissance began in Europe only after the Indian notation was adopted. Indeed the decimal notation is the very pillar of all modern civilization.” (Amartya Kumar Datta, Resonance, April 2002.) 0,1,2,3,4,5,6,7,8,9, and ‘ . ‘ (Zero and the decimal place value system)
  • Some well known Indian mathematicians
  • Aryabhata (499 AD)
  • Aryabhata (499 A.D.)  His main work: Aryabhatia, written when he was 23.  The earth is round and rotates on its axis, and the earth revolves around the sun. Rotation of earth: 23h, 56m, 4.1s (Encyclopedia)  Put forth the true scientific cause of eclipses (Encyclopedia)  The moon reflects light from the sun.  Astronomical findings were based on accurate astronomical observations. (Encyclopaedia) Main astronomical findings:
  •  “Add 4 to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.”  The Aryabhata algorithm, further developed later by Bhaskara in 621, is the standard method for solving first order Diaphontine equations (ax+by=c). Also known as Kuttaka (pulverizer) algorithm. Aryabhata (499 A.D.) 1416.3 20000 62832 
  •  “Add 4 to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.”  The Aryabhata algorithm, further developed later by Bhaskara in 621, is the standard method for solving first order Diaphontine equations (ax+by=c). Also known as Kuttaka (pulverizer) algorithm. Aryabhata (499 A.D.)
  •  “Add 4 to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.”  The Aryabhata algorithm, further developed later by Bhaskara in 621, is the standard method for solving first order Diaphontine equations (ax+by=c). Also known as Kuttaka (pulverizer) algorithm.  Modern methods for finding square roots and cube roots, these methods were introduced in Europe only in the 16th century AD. (Amartya Kumar Datta, Resonance, April 2002.) Aryabhata (499 A.D.) irrationality!
  • http://en.wikipedia.org/wiki/Aryabhata How the terms sine and cosine originated
  • Brahmagupta (598-670 AD)
  • Brahmagupta (598 - 670 A.D.)  Important works: BrAhmasphuTasiddhAnta (revised and criticized the works of earlier astronomers such as Aryabhatta) and KhanDa-KhAdyaka (astronomy and mathematics).  Discovered what is known today as Brahmagupta’s lemma for solving the so-called Pell’s equation Dx2 1 = y2, (Brahmagupta’s lemma) and solved it for D = 83 and 92. (A complete solution for any D was later provided by Jayadeva and Bhaskaracharya II through the chakravala method.)  “Symmetric formula for area of a cyclic quadrilateral, appearing for the first time in the history of mathematics” (see for e.g. ‘A modern introduction to ancient Indian mathematics’ by T. S. Bhanu Murthy, ), and expressions for the diagonals of a cyclic quadrilateral.  “Gave a simple rule for forming a ‘Brahmagupta quadrilateral’, a cyclic quadrilateral whose sides and diagonals are integral and whose diagonals intersect orthogonally.” Reference: Encyclopedia 
  • http://en.wikipedia.org/wiki/Cyclic_quadrilateral#Brahmagupta_quadrilaterals Brahmagupta (598 - 670 A.D.)
  • Brahmagupta’s results are mentioned in, for e.g., “Geometry revisited” by Coxeter and Greitzer, published by the Mathematical Association of America (MAA): Brahmagupta (598 - 670 A.D.)
  • some extracts from the book (pp. 56-59)
  •  An interpolation formula equivalent to the modern Newton-Stirling interpolation formula of second order.  Applied this method to find the sines of intermediate angles from a given table of sines.  The need to make complex calculations and the ability due to a superior number system, thus brought advances in numerical techniques as well Brahmagupta (598 - 670 A.D.) References: Encyclopedia, TAGS
  • Bhaskara I (600-680 AD)
  • Bhaskara I (600-680 AD)  Important works – MahAbhAskarIya and LaghubhAskarIya (provided explanations and interpretations of Aryabhata’s reasonings).  AryabhatIyabhAshya – a commentary on Aryabhatia (dated 628 AD.).  Provided a compact classification of mathematics into different specializations (Encyclopedia).  Responsible for evolving trigonometry in its present form (ardhajya etc. see encyclopedia), and created the modern trigonometric circle.  Gave an approximation for the sine.  Elaborated on the kuttaka method of Aryabhata.
  • Bhaskara II (Bhaskaracharya) (1114 AD)
  • Bhaskara II (AD. 1114) • Several treatises – Lilavati, a standard work of Hindu mathematics, covering arithmetic, algebra, geometry and mensuration. Many eminent Sanskrit mathematicians wrote commentaries (bhashyas) on it. • Bijaganita – standard treatise on Hindu algebra. More advanced text than the above. • Siddhanta shiromani – standard treatise on Hindu astronomy.
  •  Developed a general algorithm (the chakravala algorithm, based on Jayadeva’s earlier work of 11th cent.) to obtain integral solutions to the so- called Pell’s equation: 22 1 yDx  where D is also an integer.  Can be used to fond solutions for any D. D=61 and 109 are especially difficult, but Bhaskara used the chakravala algorithm to find the solution in a few lines! Solution for D=61, x = 226, 153, 980 and y = 1, 766, 319, 049 Solution for D=109, x = 15140424455100 and y = 158070671986249 In 1657 Fermat (unaware of the chakravala method) proposed the above equation with D = 61 to Frénicle as a challenge problem. Bhaskara II (AD. 1114) Encyclopedia
  •  Developed a general algorithm (the chakravala algorithm, based on Jayadeva’s earlier work of 11th cent.) to obtain integral solutions to the so- called Pell’s equation: 22 1 yDx  where D is also an integer.  Can be used to fond solutions for any D. D=61 and 109 are especially difficult, but Bhaskara used the chakravala algorithm to find the solution in a few lines! Solution for D=61, x = 226, 153, 980 and y = 1, 766, 319, 049 Solution for D=109, x = 15140424455100 and y = 158070671986249 In 1657 Fermat (unaware of the chakravala method) proposed the above equation with D = 61 to Frénicle as a challenge problem. Bhaskara II (AD. 1114) Encyclopedi “What would have been Fermat’s astonishment if some missionary, just back from India, had told him that his problem had been successfully tackled there by native mathematicians almost six centuries earlier?” -André Weil, in “Number Theory, an approach through history from Hammurapi to Legendre” (pp. 81-82)
  •  Developed a general algorithm (the chakravala algorithm, based on Jayadeva’s earlier work of 11th cent.) to obtain integral solutions to the so- called Pell’s equation: 22 1 yDx  where D is also an integer.  Can be used to fond solutions for any D. D=61 and 109 are especially difficult, but Bhaskara used the chakravala algorithm to find the solution in a few lines! Solution for D=61, x = 226, 153, 980 and y = 1, 766, 319, 049 Solution for D=109, x = 15140424455100 and y = 158070671986249 In 1657 Fermat (unaware of the chakravala method) proposed the above equation with D = 61 to Frénicle as a challenge problem. Bhaskara II (AD. 1114) “The chakravala method anticipated the european methods by more than a thousand years. But no European performances in the whole field of algebra at a time much later than Bhaskara’s, nay nearly up to our times, equalled the marvellous complexity and ingenuity of chakravala.” - Selenius “What would have been Fermat’s astonishment if some missionary, just back from India, had told him that his problem had been successfully tackled there by native mathematicians almost six centuries earlier?” -André Weil, in “Number Theory, an approach through history from Hammurapi to Legendre” (pp. 81-82) Encyclopedi
  • Bhaskara II (AD. 1114)  Laid the seeds of differential calculus as shown, for e.g., the following formula as well as its geometrical demonstration, for calculating the instantaneous velocity (tatkalika gati) of planets  dd  cos)(sin (which is a result of differential calculus arrived at by taking limits.)  Knew that the derivative vanishes at the points of extrema.  Discovered what is known today as Rolle’s theorem in mathematical analysis/calculus. Encyclopedia The seeds of differential calculus:
  • Bhaskara II (AD. 1114) 1. All the arithmetic (except decimal representation) taught in school today is described in Lilavati - addition, subtraction, and division, multiplication, finding square roots and cube roots for integers. 2. All the rules taught in school for manipulating fractions: addition, subtraction, multiplication, and division of fractions, as well as finding square roots and cube roots of fractions are also clearly defined and described. 3. The basic algebra taught in school – that of representing an unknown quantity by a symbol and setting up an algebraic equation and solving it to find the value of that quantity, is described in Lilavati in full detail. In fact the chapter dealing with this seems to be straight out of a modern school text book! 4. Explicit rules on handling zero and a clear notion of the limit which forms the base of infinitesimal calculus. 5. A section on quadratic equations containing the standard method of solving such equations by completing the square. 6. All the results on arithmetic progressions, geometric progressions, permutations and combinations which are taught in high school are clearly described in Lilavati. 7. Other interesting results such as the sum of the first n whole numbers, the sum of the squares of the first n whole numbers, and the sum of the cubes of the first n whole numbers are also presented. From Lilavati:
  • A few observations • What is discussed is just a tip of the iceberg. • It shows a thriving and vibrant scientific culture in India, open to criticizing and evolving and building on the works of earlier scientists. It was open to criticize and modify earlier works and was not stuck in dogmas - much like modern science is (supposed to be) practiced today. • Presents a picture opposite to that depicted in the mainstream historical accounts as India being a backward and stagnant civilization. • Seriously questions the belief that the Greeks were the only mathematicians worth mentioning – this is a view for which there is no proof, only the repeated claims of Eurocentric scholars repeating the Greek – Renaissance sequence.
  • However… • The conventional belief is that the Greeks invented their philosophy and mathematics by themselves, without external influence. • Even though India was at the time well known for her scientific, mathematical, artistic and philosophical knowledge, • And even though several Greeks visited India to acquire Indian knowledge, and there are, as a result, several similarities between Greek and Indian philosophies.
  • some points to note • Greek knowledge did not originate in a vacuum. • There were several visits by Greeks to India and similarities in philosophy. • We now mention several instances where the Greeks travelled to India and were influenced by the Indian knowledge system and incorporated it into their own.
  • Greek visits to India • Darius I sent Skylax to explore the Indus in 519 BC. Later called frequent meetings between Greeks and Indians for counsel and discussion. (TAGS) • Aristoxenes (350-300 BC) mentions a dialogue between Socrates and an Indian philosopher. (TAGS)
  • Greek visits to India • Key Greek philosophers such as Plato, Democritus, Pherecydes, and Pythagoras, are known to have travelled to India.
  • • Key Greek philosophers such as Plato, Democritus, Pherecydes, and Pythagoras, are known to have travelled to India. Greek visits to India “We know that he (Pythagoras) went to India to be instructed” -Reuben Burrow, Hints concerning the observatory at Benares, 1783
  • • Key Greek philosophers such as Plato, Democritus, Pherecydes, and Pythagoras, are known to have travelled to India. Greek visits to India “We know that he (Pythagoras) went to India to be instructed” -Reuben Burrow, Hints concerning the observatory at Benares, 1783 -William Hamilton, The history of medicine, surgery, and anatomy, Vol. I (1831)
  • • Key Greek philosophers such as Plato, Democritus, Pherecydes, and Pythagoras, are known to have travelled to India. Greek visits to India “We know that he (Pythagoras) went to India to be instructed” -Reuben Burrow, Hints concerning the observatory at Benares, 1783 -William Hamilton, The history of medicine, surgery, and anatomy, Vol. I (1831) “Journeys to India and indebtedness to Brahminical wisdom are now ascribed to numerous founders and leaders in Greek thought, such as Plato, Democritus, Pherecydes of Syrus and, quite often, Pythagoras.” -Wilhelm Halbfass, India and Europe: an essay in understanding, Albany, State University of New York Press (1988), p. 16.
  • Greek visits to India • Several Greek philosophers travelled with Alexander when he invaded India and interacted with Indian sages. These include Onesicritius, Cynic, Democritean, Anaxagoras, and Pyrhho. • Pali Buddhist literature records religious and philosophical dialogues between the Buddhist monk Nagasena and the Indo-Greek ruler Menander. • Gnostic philsopher Bardesanes of Edessa (ca AD 200) travelled to India. • The founder of the Neoplatonic school, Plotinus, went to India in AD 242 expressly to study its philosophy.
  • Greek visits to India • Several Greek philosophers travelled with Alexander when he invaded India and interacted with Indian sages. These include Onesicritius, Cynic, Democritean, Anaxagoras, and Pyrhho. • Pali Buddhist literature records religious and philosophical dialogues between the Buddhist monk Nagasena and the Indo-Greek ruler Menander. • Gnostic philsopher Bardesanes of Edessa (ca AD 200) travelled to India. • The founder of the Neoplatonic school, Plotinus, went to India in AD 242 expressly to study its philosophy. O. P. Jaggi, Indian System of Medicine, Vol. 4 of History of Science and Technology of India, Delhi, Atma Ram and sons, 1973 Amiya Kumar Roy Chowdhury, Man, Malady, and Medicine – History of Indian Medicine, Calcutta, Das Gupta and Co. Ltd, 1988 TAGS
  • Similarities in philosophies • Pythagoras and his followers believed in the transmigration of the soul (reincarnation) – a typically Indian (Hindu/Buddhist) concept. He himself claimed having fought in the Trojan war in a previous incarnation. • The Pythagoreans also were strict vegetarians, again a trait typical to Hindus/Buddhists/Jains.
  • Similarities in philosophies • Pythagoras and his followers believed in the transmigration of the soul (reincarnation) – a typically Indian (Hindu/Buddhist) concept. He himself claimed having fought in the Trojan war in a previous incarnation. • The Pythagoreans also were strict vegetarians, again a trait typical to Hindus/Buddhists/Jains. H. G. Rawlinson, “Early contacts between India and Europe”, in A Cultural History of India, A. L. Basham (ed.) (Oxford University Press, 1975) (p. 427-428)
  • Similarities in philosophies • The concept of karma is essential to Plato’s philosophy.
  • Similarities in philosophies • The concept of karma is essential in Plato’s philosophy. “Metempsychosis, with the complementary doctrine of karma, is the key- stone of the philosophy of Plato. The soul is for ever travelling through a ‘cycle of necessity’: the evil it does in one semicircle of its pilgrimage is expiated in the other. ‘Each soul’, we are told in the Phaedrus, returning to the election of a second life, shall receive one agreeable to his desire.’ “ “…most striking of all is the famous apologue of Er the Pamphylian, with which Plato appropriately ends the Republic. … ‘In like manner, some of the animals passed into men, and into one another, the unjust passing into the wild, and the just into the tame.” -H. G. Rawlinson, “Early contacts between India and Europe”, in A Cultural History of India, A. L. Basham (ed.) (Oxford University Press, 1975) (p. 427-428)
  • Similarities in philosophies • The theory that matter consists of four elements (earth, water, air and fire) was taught by Empedocles (490-430 BC), disciple of Pythagoras. The later Aristotelian description of the physical world included the ether (space) element as well. • Indian philosophy also describes the physical world in terms of these five elements.
  • Similarities in philosophies • The theory that matter consists of four elements (earth, water, air and fire) was taught by Empedocles (490-430 BC), disciple of Pythagoras. The later Aristotelian description of the physical world included the ether (space) element as well. • Indian philosophy also describes the physical world in terms of these five elements. (Bhagavad Gita 7.4 and 7.5.) Rough translation - Earth, water, fire, air, and ether, (describes space and matter) as well as mind, intellect, and ego (describes consciousness) , are My apara (‘lower’) nature (which keep on changing with time), while the para (‘higher’) nature is the unchangeable Self (Atman), which being beyond Time, is beyond change as well.
  • Similarities in philosophies • Many similarities between Greek medicine and Indian medicine. • Pythagoras is known to have travelled to India and upon his return to have influenced the Hippocratic system of medicine. • The Hippocratic collection mentions an Indian regime for cleaning the teeth, as well as listing drugs of Indian origin, some with corrupted Sanskrit names.
  • Some observations… • In Indian civilization, science (apara vidya), unlike the materialistic world view of today, is not opposed to the spiritual quest (para vidya). • And the spiritual quest (what is mistranslated as religion in today’s context) is not opposed to the pursuit of material science, and is not a set of dogmas to be blindly adhered to. • In fact both spirituality (para vidya) and science (apara vidya) are the two sides of the same coin, (‘the coin of wisdom’, ‘the coin of knowledge’). None of the sides are ignored at the others’ expense.
  • Some observations… • In Indian civilization, science (apara vidya), unlike the materialistic world view of today, is not opposed to the spiritual quest (para vidya). • And the spiritual quest (what is mistranslated as religion in today’s context) is not opposed to the pursuit of material science, and is not a set of dogmas to be blindly adhered to. • In fact both spirituality (para vidya) and science (apara vidya) are the two sides of the same coin, (‘the coin of wisdom’, ‘the coin of knowledge’). None of the sides are ignored at the others’ expense. “…the Vedic Hindu, in his great quest of the para vidya (absolute truth), made progress in the apara (relative truth), including the various arts and sciences, to a considerable extent, and with a completeness which is unparalleled in antiquity.” –Bibhutibhushan Datta, Ancient Hindu Geometry, 1993. “…the culture of the science of mathematics or of any other branch of secular knowledge, was not considered to be a hindrance to spiritual knowledge. In fact, apara vidya was then considered to be a helpful adjunct to para vidya.” -B. Datta and A. N. Singh, History of Hindu Mathematics, 1962.
  • Genius down south: The Kerala school (1300-1600 AD)
  • Kerala school (1300-1600) • Pioneered by Madhava of Sangamagrama (1340-1425) (Today Irinjalakuda in Thrissur district). • Continues and develops upon the findings of the Aryabhata school. • The mathematicians and astronomers of this school formed a continuous line till the 17th century and made several important contributions to calculus, trigonometry, spherical trigonometry and astronomy. • Most of Madhava’s original writings are lost, but his work survives in the bhashyas (commentaries) by later scholars of the school.
  • Kerala school (1300-1600) • Vatasseri Parameshvara Nambudiri (1380- 1460) • Damodara • Nilakantha Somayaji (1444-1544) • Jyeshthadeva (1500-1610) • Achyuta Pisharati (1550-1621) • Melpathur Narayana Bhattathiri (1559-1645) • Sankara Varma (1800-1838)
  • Kerala school (1300-1600) • Tantra Sangraha in 1501 and Aryabhatia-Bhashya by Nilakantha Somayaji (Sanskrit). • Yuktibhasha by Jyeshthadeva in 1530 (Malayalam). Elaborates further on the Tantra Sangraha. • Kriyakramakari (a commentary on Bhskaracharya’s Lilavati) andYukti-Dipika (commentary on Tantrsangraha) by Shankara Variyar (1500-1550). • Sadratnamala by Shankaravarman in 1819 (Sanskrit). • Karana Paddhati • These books are commentaries on the results of Madhava and contain several new results developing on his work.
  • Charles Whish (1794-1833), (civil servant for the East India Company) Transactions of the Royal Asiatic Society of Great Britain and Ireland, Vol. 3, No. 3, (1834)
  • Kerala school (1300-1600): Key discoveries • Irrationality of π From the Aryabhatia of Aryabhatia “Add 4 to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.” (i.e. it is an approximation.)
  • Kerala school (1300-1600): Key discoveries • Irrationality of π From the Aryabhatia of Aryabhatia “Add 4 to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.” (i.e. it is an approximation.) Nilakantha in Aryabhatia-bhashya: “Why has an approximate value been mentioned here (in Aryabhatia) instead of the actual value?” And goes on to give the answer: “Given a certain unit of measurement in terms of which the diameter specified has no fractional part, the same measure when employed to specify the circumference will certainly have a fractional part…even if you go on a long way (i.e. keep on reducing the measure of the unit employed), the fractional part will only become very small. A situation in which there will be no fractional part is impossible, and this is what is the import of the expression Asanna (can be aproached).” -Development of calculus in India: contribution of kerala school (1350-1550 CE), K. Ramasubramanian, IIT Bombay
  • Kerala school (1300-1600): Key discoveries • Infinite geometric series Jyeshthadeva’s proof in Yuktibhasha:                              x xx x xx x x x 1 1 1 1 1 11 1 1 1 1 1 2          x x xxx n n 1 1 1 2          x xxx 1 1 1 32  for |x|<1 nlimitthetakeNow
  • Kerala school (1300-1600): Key discoveries • Series expansions for sin, cos and arctan functions along with the error after n terms • An approximate proof for the arctan series in modern notation follows  !7!5!3 sin 753 xxx xx  !6!4!2 1cos 642 xxx x  753 tan 753 1 xxx xx
  • Kerala school (1300-1600): Key discoveries    642 2 1 1 1 1 tan xxx x x dx d Integrating both sides from 0 to x, we get  753 tan 753 1 xxx xx Derivation of series for arctan:
  • Kerala school (1300-1600): Key discoveries    642 2 1 1 1 1 tan xxx x x dx d Integrating both sides from 0 to x, we get  753 tan 753 1 xxx xx Derivation of series for arctan: Putting x = 1:
  • Kerala school (1300-1600): Key discoveries    642 2 1 1 1 1 tan xxx x x dx d Integrating both sides from 0 to x, we get  753 tan 753 1 xxx xx Derivation of series for arctan: Putting x = 1: Putting x = 1: 7 1 5 1 3 1 1 4 
  • Kerala school (1300-1600): Key discoveries    642 2 1 1 1 1 tan xxx x x dx d Integrating both sides from 0 to x, we get  753 tan 753 1 xxx xx Derivation of series for arctan: Putting x = 1: Putting x = 1: 7 1 5 1 3 1 1 4  The so-called Gregory series, published by Gregory in 1668, but discovered by Madhava 300 years earlier!
  • Kerala school (1300-1600): Key discoveries • Not just the series, the error after retaining n terms was also obtained. • The Madhava (Gregory) series converges very slowly: the first 200 terms add up to one- fourth of 3.1466. • Nilakantha in Tantra Sangraha used the knowledge of the error to obtain series with much faster convergence.
  • Kerala school (1300-1600): Key discoveries • Not just the series, the error after retaining n terms was also obtained. • The Madhava (Gregory) series converges very slowly: the first 200 terms add up to one- fourth of 3.1466. • Nilakantha in Tantra Sangraha used the knowledge of the error to obtain series with much faster convergence.        77 1 55 1 33 1 4 1 1 4 333 
  • Kerala school (1300-1600): Key discoveries • Not just the series, the error after retaining n terms was also obtained. • The Madhava (Gregory) series converges very slowly: the first 200 terms add up to one- fourth of 3.1466. • Nilakantha in Tantra Sangraha used the knowledge of the error to obtain series with much faster convergence.        77 1 55 1 33 1 4 1 1 4 333         545 4 343 4 141 4 4 555 
  • Kerala school (1300-1600): Key discoveries • Not just the series, the error after retaining n terms was also obtained. • The Madhava (Gregory) series converges very slowly: the first 200 terms add up to one- fourth of 3.1466. • Nilakantha in Tantra Sangraha used the knowledge of the error to obtain series with much faster convergence.        77 1 55 1 33 1 4 1 1 4 333         545 4 343 4 141 4 4 555  and several others…
  • Verses from Sadratnamala  753 tan 753 1 xxx xx
  • Verses from Sadratnamala value of π is calculated to 17 decimal places and expressed by the famous kattapayadi system
  • Kerala school (1300-1600): Key discoveries • Several discoveries in spherical trigonometry and astronomy: an early planetary model which was identical to the one proposed by Tycho Brahe. • Several of their works are still subjects of research by modern mathematicians (much like Srinivasan Ramanujan’s works)!
  • A case for the possible transmission of the mathematics and astronomy of the Kerala school to Europe
  • Historical background • In the middle ages Europe is centuries behind India in mathematical knowledge. • In 1202 the Indian number system is popularized in Europe by Fibonacci. • At the same time, Europe is engulfed in the Dark ages and abject poverty. Hence trade and conquest with wealthy nations such as India assumes importance. • But trade implies navigating the seas, which needs knowledge of astronomy, the ability to calculate latitude and longitude, which in turn requires knowledge of trigonometry, tables of sines etc. Also, a reliable calendar is a must. • Neither did Europe have the knowledge, nor was its calendar reliable enough for navigation. Thus several ships were lost accompanied by severe economic and human losses. • Thus navigation and calendar reform become priority programs by the church. Lucrative prizes are offered for anyone who could provide accurate techniques.
  • Historical background • At the same time, Indian mathematics had all this information. Indian navigators used to do trade with several countries. Thus it became important to acquire this knowledge. • But Hindus were ‘pagans’, ‘heathens’, and ‘idol worshippers’ who had to be ‘civilized’ (christianized). So though it privately sought ‘pagan’ learning, publicly it continued to deny that there was any learning among the ‘pagans’. • Anyone who acknowledged ‘pagan’ sources of knowledge would be burnt at the stake for being a heretic. Thus, although ‘pagan’ knowledge was appropriated (as shall be seen), the sources were never acknowledged.
  • Historical background • At the same time, Indian mathematics had all this information. Indian navigators used to do trade with several countries. Thus it became important to acquire this knowledge. • But Hindus were ‘pagans’, ‘heathens’, and ‘idol worshippers’ who had to be ‘civilized’ (christianized). So though it privately sought ‘pagan’ learning, publicly it continued to deny that there was any learning among the ‘pagans’. • Anyone who acknowledged ‘pagan’ sources of knowledge would be burnt at the stake for being a heretic.
  • Opportunity and means • 1499- Vasco da Gama arrives at the Malabar coast in Kerala and establishes a direct link to Europe via Lisbon. • 1540 – Francis Xavier arrives in Goa and makes Kerala a hub of missionary activities (missionary activity is still vigorous in Kerala ). • Jesuit mathematician and astronomer Christoph Clavius includes mathematics in the curriculum of Jesuit priests at Collegio Romano. (Clavius later headed the calendar reform committee.) • The first batch of Jesuit priests mathematically trained by Clavius reach Malabar (including the city of Cochin, the epicenter of the Kerala mathematicians) 1578 onwards. These include: Matteo Ricci, Johann Schreck, and Antonio Rubino. • It is clear that the express purpose is to acquire Indian knowledge on navigation, astronomy and the calendar (panchang). They learn the local language and are in close touch with local scholars and royal personages. • Also, Rubino and Ricci have been recorded in correspondence as answering requests for astronomical information from Kerala sources.
  • Circumstantial Evidence • 1597 - Tycho Brahe becomes the Royal astronomer of the Holy Roman empire upon the invitation of emperor Rudolph II to Prague. • In this capacity, he is a natural recipient of Indian astronomy texts obtained by Jesuit priests from Kerala. • Is it just a coincidence that his model of planetary motion, the ‘Tychonic model’, is identical to the one proposed by Nilakantha in his Tantra Sangraha in 1501? • Jyeshthadeva’s Yuktibhasha gives a formula involving a passage to infinity to calculate the area under a parabola. The same formula was used by Fermat, Pascal, and John Wallis. • The chronology of the events, and the circumstantial evidence is too strong to be a mere coincidence.
  • Circumstantial Evidence • 1597 - Tycho Brahe becomes the Royal astronomer of the Holy Roman empire upon the invitation of emperor Rudolph II to Prague. • In this capacity, he is a natural recipient of Indian astronomy texts obtained by Jesuit priests from Kerala. • Is it just a coincidence that his model of planetary motion, the ‘Tychonic model’, is identical to the one proposed by Nilakantha in his Tantra Sangraha in 1501? • Jyeshthadeva’s Yuktibhasha gives a formula involving a passage to infinity to calculate the area under a parabola. The same formula was used by Fermat, Pascal, and John Wallis. • The chronology of the events, and the circumstantial evidence is too strong to be a mere coincidence. “This very strange current-day belief that only Christians, or their theologically correct predecessors in Greece have developed almost all serious knowledge in the world demonstrates the strength of the continuing cultural feeling against ‘pagan’ learning. There is nothing ‘natural’ or universal in hiding what one has learnt from others: the Arabs, for instance, did not mind learning from others, and they openly acknowledged it. This is another feature unique to the church: the idea that learning from others is something so shameful that, if it had to be done, the fact ought to be hidden. Therefore, though the church sought knowledge about the calendar, specifically from India, and profusely imported astronomical texts,…this import of knowledge remained hidden.” -D. P. Agrawal References: (1) D. F. Almeida and G. G. Joseph, Eurocentrism in the history of mathematics: the case of the Kerala school, Race and Class (2004). (2) Cultural foundations of mathematics, C. K. Raju
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