8/04/2006

Ancient Indian Mathematics

"I have never found one among them who could deny that a single shelf of a good European library was worth the whole native literature of India and Arabia. "



India was usually known to outsiders until very recently as the land of naked sadhus, snake charmers, overtly religious mystical land which taught the world how to make love. Such a perspective definitely helped the colonisers in assigning an inferior status for the colonised and justify their acts of colonisation and "civilisation". Macaulay's remarks in his Minute on Indian Education is a good read to understand the British perspective then about India. Many influential political leaders (Gandhi for example) who had renounced the more superstitious religious beliefs, in favour of less superstitious beliefs also contributed to the mainstream western perspective on India remaining a status quo. This is surprising considering the fact that even during Gandhi's time, there were huge difference of opinion between Gandhi and other more rational minded people like Tagore and Ambedkar. Gandhi in his essay Hind Swaraj, strongly condemds technology and industrialization, whereas Tagore believed in science, industry and rationality. Tagore was particularly disgusted by Gandhi remark that the 1934 Bihar earthquake was a divine retribution for ill treatment of dalits. Tagore, who was as much against caste exploitation as Gandhi was, however could not stand such an irrational explanation for the earthquake suggested by Gandhi. Their debates on this and many other issues (like Gandhi's belief that everyong should use the carka for atleast 30 mins a day) in the form of letters make a very good read [1]. Another non ignorable figure who had serious issues with Gandhi's irrationality was Ambedkar. The latter firmly believed in the evils of caste system, where as Gandhi who was against the opression against dalits, nevertheless believed in the sanctitty of the age old caste system and even justified its working as a means to achieving social stability. You can read Ambedkars response to Gandhi's views on caste in his essay "Annihilation of Caste". It looks like Tagore and Ambedkar were not at all influential in changing outsiders outlook towards India. Gandhi definitely takes the spotlight.

Gladly enough, thanks to the IT age, outsourcing and the recent economic boom, this image is rapidly changing. Such a categorization, not only misrepresented India, but also ignored completely many other aspects like agnosticism (Buddhism), atheism (Carvaka) and science and mathematics which were very much being developed in ancient India. Buddhism, which originated in India and spread as a rebel against caste system and rituals, was in full glory for about a millenium and was one of the prime exports of India during those days to China and other eastern countries. So much that China refered to India as the Kingdom of Buddhism. Rationalism and skepticism too, was a school of thought which cannot be ignored during these times. Hinduism, Buddhism, Jainism, Atheism and many other schools were competing at a time for acceptance as a mainstream philosophy. Amartya Sen's book The Argumentatvie India forms an excellent read on this subject.

As far as ancient Indian sciences are concerned, it was developed and transmitted to regions outside India by Chinese and Arabs. Many mathematics and science texts were translated by Al-Beruni[2], Al-Khwarizmi (who is responsible for the temrs algebra and algorithm) into Arabic and found its way to Europe there on. One of the prime motivations for doing mathematics those days was astronomy. Approximations for trigonometric values, pi, circumference of circle, etc were developed. Bhaskara's approximation for sin(x) as a rational function of x, Brahmagupta's calculations of eclipses, Aryabhatta I's approximation of pi, Aryabhatta I's method to solve linear Diophantine equations are just some examples amongst a huge literature.

Broadly, ancient Indian mathematics can be categorized into the following based on their period of development.

  • Pre Vedic Indus mathematics dating back to as early as 3000BC. Development of mathematics were highly influenced by practical applications like measuring scales, calculating brick ratios etc.
  • Vedic or Sulbasutras, which contained rules to construct altars for various rites and rituals. Various constructions based on Pythagoras theorem are listed, approximation to square root of 2, approximation to pi are some of the achievements. This period probably lasted till around 500BC.
  • Jaina mathematics, from 600BC to 500AD. Prime achievements are various notions of infinities, Pascal's triangle, form of set theory, operations with roots of order larger than 2 etc
  • A set of manuscripts were found around 1880s, called the Bakhshali manuscript. The dates of these manuscripts are assumed to be around 400AD. This book contains sets of problems and solutions in linear equations, fractions, square roots etc
  • Golden age of Indian mathematics set off by Aryabhatta I. Numbers start becoming more abstract and makes it possible to consider zero and negative numbers. Brahmagupta, Mahavira and Bhaskara lists down rules for multiplying, subtracting and (wrongly) dividing by zero. All of them missed that dividing by zero does not make sense :) It was during this period that Al-Khwarizmi, who lived from 790 to 840, wrote Al'Khwarizmi on the Hindu Art of Reckoning and passed on Indian achievements to Islamic and Arabic countries. Ibn Ezra, who lived from 1092 to 1167 in Spain also helped transmit Indian number system to Europe.
  • Keralese mathematics: The golden age started to decline after Bhaskara's works around 1200. There is some consensus that political instabilities are partly to be blamed for this decline. Despite all these, mathematics continued to develop in Kerala. The time period between 1400 and 1600 is considered to be the peak period of this development. In addition to building upon and extending previous works by Bhaskara, Brahmagupta, Aryabhatta etc, one of the significant achievements of this period was mathematical inductive proof in the works of Nilakantha, Jyeshtadeva. The prominent figure during this time period is definitely Madhava of Sangamagramma. He built tools essential for modern analysis. Though not fully certain, he is supposed to have independently derived taylor series expansion for arctan, sin, cos and many other function. A list of 13 different expansions attributed to Madhava is listed here. Madhava also managed to calculate the value of pi to 17 decimal places (3.14155265358979324), much ahead of his contemporaries.

People have started recognizing and acknowledging role of Indians, but awareness still seems to be a big issue. I think it will take some time before terms like Madhava-Gregory series, Leibniz-Gregory-Madhava constant, Hemachandra-Fibonacci series, Bhaskara-Brouncker Algorithm, Aryabhatta algorithm, Aryabhatta Remainder Theorem, Aryabhatta Kuttaka etc become commonplace.You might want to check the following links to dig deeper into the history of Indian mathematics.

    1. Indian Mathematics: Redressing the balance
    2. C. T. Rajagopal and M. S. Rangachari. 'On an untapped source of medieval Keralese mathematics', Archive for History of Exact Sciences 18 (pages 89-102). 1978.
    3. Keralese Mathematics
    4. Possible transmission of Keralese mathematics to Europe.


    Here is a list of prominent Indian mathematicians in chronological order. The names and brief description are obtained from various websites including but not limited to University of St. Andrews.



    Baudhayana 800BC - 740BC

    Baudhayana was the author of one of the earliest Sulbasutras: documents containing some of the earliest Indian mathematics.



    Apastamba 600BC - 540BC

    Apastamba was the author of one of the most interesting of the Indian Sulbasutras from a mathematical point of view.



    Panini 520BC - 460BC

    Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology.



    Katyayana
    200BC - 140BC

    Katyayana was the author of one of the Sulbasutras: documents containing some of the earliest Indian mathematics.



    Yavanesvara
    120 - 180

    Yavanesvara was an Indian astrologer who translated an important Greek text on astrology.



    Aryabhata the Elder 476 - 550

    Aryabhata I was an Indian mathematician who wrote the Aryabhatiya which summarises Hindu mathematics up to that 6th Century.



    Yativrsabha 500 - 570

    Yativrsabha was a Jaina mathematician who gave a description of the universe which is of historical importance in understanding Jaina science and mathematics.



    Varahamihira 505 - 587

    Varahamihira was an Indian astrologer whose main work was a treatise on mathematical astronomy which summarised earlier astronomical treatises. He discovered a version of Pascal's triangle and worked on magic squares.



    Brahmagupta 598 - 670

    Brahmagupta was the foremost Indian mathematician of his time. He made advances in astronomy and most importantly in number systems including algorithms for square roots and the solution of quadratic equations.



    Bhaskara I 600 - 680

    Bhaskara I was an Indian mathematicians who wrote commentaries on the work of Aryabhata I.



    Lalla 720 - 790

    Lalla was an Indian mathematician who wrote mainly on the application of mathematics to astronomy.



    Govindasvami 800 - 860

    Govindasvami was an Indian mathematical astronomer whose most famous treatise was a
    commentary on work of Bhaskara I.




    Mahavira 800 - 870

    Mahavira Mahavira was an Indian mathematician who extended the mathematics of Brahmagupta.


    Prthudakasvami 830 - 890

    Prthudakasvami was an Indian mathematician best known for his work on solving equations.



    Sankara Narayana
    840 - 900

    Sankara Narayana was an Indian astronomer and mathematician. He wrote a commentary on the work of Bhaskara I.


    Sridhara 870 - 930

    Sridhara was an Indian mathematician who wrote on practical applications of algebra and was one of the first to give a formula for solving quadratic equations.


    Aryabhata II 920 - 1000

    Aryabhata II was an Indian mathematician who wrote about astronomy as well as geometry. He constructed tables of sines accurate up to about 5 figures.



    Vijayanandi 940 - 1010

    Vijayanandi was an Indian mathematician and astronomer who made some contributions to trigonometry.


    Sripati 1019 - 1066

    Sripati was an Indian who wrote works on astronomy and arithmetic.


    Brahmadeva 1060 - 1130

    Brahmadeva was an Indian mathematician who wrote a commentary on the work of Aryabhata I.


    Acharya Hemchandra 1089 - 1173

    Hemachandra was a Jaina scholar who presented what is now called the Fibonacci sequence around 1150, about 50 years before Fibonacci (1202). He was considering the number of cadences of length n, and showed that these could be formed by adding a short syllable to a cadence of length (n−1), or a long syllable to one of (n−2). This recursion relation F(n) = F(n−1) + F(n−2) is what defines the fibonacci sequence.
    Bhaskara 1114 - 1185

    Bhaskara II or Bhaskaracharya was an Indian mathematician and astronomer who extended Brahmagupta's work on number systems.


    Narayana Pandit 1340-1400

    Narayana Pandit is the author of an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati).


    Madhava of Sangamagrama 1340-1425

    Madhava of Sangamagrama was the founder of the Kerala School and considered to be one of the greatest mathematician-astronomers of the Middle Ages. It is vaguely possible that he may have written Karana Paddhati a work written sometime between 1375 and 1475 but all that is known of Madhava comes from works of later scholars.


    Parameshvara 1370-1460

    Parameshvara, the founder of the Drgganita system of astronomy, was a prolific author of several important works. He belonged to the Alathur village situated on the bank of Bharathappuzha. He is stated to have made direct astronomical observations for fifty-five years before writing his famous work, Drgganita. He also wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His Lilavati Bhasya, a commentary on Bhaskara II's Lilavati, contains one of his most important discoveries:


    Nilakantha Somayaji 1444 - 1544

    Nilakantha was a mathematician and astronomer from South India who wrote texts on both astronomy and infinite series.


    Jyesthadeva 1500 - 1575

    Jyesthadeva was a mathematician from South India who wrote an important work on mathematics and astronomy which summarises the work of the Kerala school.


    Citrabhanu c. 1530

    Citrabhanu was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:


    Kamalakara 1616 - 1700

    Kamalakara was an Indian astronomer and mathematician who combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists.


    Jagannatha Samrat 1690 - 1750

    Jagannatha was an Indian mathematician who is important as a translator of important Greek works into Sanskrit.


    Sankara Varman 1800-1838

    Sankara Varman
    There remains a final Kerala work worthy of a brief mention, Sadratnamala an astronomical treatise written by Sankara Varman serves as a summary of most of the results of the Kerala School. What is of most interest is that it was composed in the early 19th century and the author stands as the last notable name in Keralese mathematics. A notable contribution was his compution of π correct to 17 decimal places.



    [1]

    The Mahatma and the Poet : Letters and Debates between Gandhi and Tagore 1915-1941, Compiled and Edited by Sabyasachi Bhattacharya, National Book Trust, India


    [2]

    Alberuni's India, Edward C. Sachau, Trobner & Co., London, 1888, Rupa & Co., 2002. You can read a review of this book at hindu.com or obtain a pdf file of his book from infinityfoundation.com

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