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Ganita Shastra (गणितशास्त्रम्) (view source)
Revision as of 18:30, 24 March 2020
, 18:30, 24 March 2020Commentaries - added content to be edited
== Introduction ==
== Introduction ==
Ganita or Indian mathematics is quintessentially a science of computation and texts of Indian mathematics essentially present systematic and efficient procedures or algorithms for the solution of various mathematical problems. The ancient texts of geometry, Shulbasutras (of the [[Kalpa Vedanga (कल्पवेदाङ्गम्)|Kalpa Vedanga]]), give us procedures for the construction and transformation of geometrical figures. The much later classical text Aryabhatiya of Aryabhata presents most of the procedures of arithmetic, algebra, geometry and trigonometry, which are taught today in schools, in just thirty-two verses comprising the Ganitapada.<ref>M. D. Srinivas,"''[http://iks.iitgn.ac.in/wp-content/uploads/2016/02/On-the-Nature-of-Mathematics-and-Scientific-Knowledge-in-Indian-Tradition-MD-Srinivas-2016.pdf On the Nature of Mathematics and Scientific Knowledge in Indian Tradition]''" Chennai: Centre for Policy Studies</ref>
Ganita or Indian mathematics is quintessentially a science of computation and texts of Indian mathematics essentially present systematic and efficient procedures or algorithms for the solution of various mathematical problems. The ancient texts of geometry, Shulbasutras (of the [[Kalpa Vedanga (कल्पवेदाङ्गम्)|Kalpa Vedanga]]), give us procedures for the construction and transformation of geometrical figures. The much later classical text Aryabhatiya of Aryabhata presents most of the procedures of arithmetic, algebra, geometry and trigonometry, which are taught today in schools, in just thirty-two verses comprising the Ganitapada.<ref name=":1">M. D. Srinivas,"''[http://iks.iitgn.ac.in/wp-content/uploads/2016/02/On-the-Nature-of-Mathematics-and-Scientific-Knowledge-in-Indian-Tradition-MD-Srinivas-2016.pdf On the Nature of Mathematics and Scientific Knowledge in Indian Tradition]''" Chennai: Centre for Policy Studies</ref>
Chandogya Upanishad's [[Narada Sanatkumara Samvada (नारदसनत्कुमारयोः संवादः)|Narada Sanathkumara Samvada]], clearly elucidates the existence of the subjects of sciences and arts depicting their antiquity in ancient India. Narada, entreated by Sanathkumara, enumerates the various sciences and arts studied by him and this list includes astronomy (nakshatra-vidya) and arithmetic (rasi-vidya). The culture of science of astronomy and mathematics classified under Aparavidya were not considered to be a hindrance to Paravidya or spiritual knowledge; they were part of the Chaturdasha and Asthadasa vidyas which was the basic curriculum of education. On the contrary they were considered as helpful adjuncts and were studied to aid the progress of Paravidya as expounded in Mundakopanishad (1.1.3-5).
Chandogya Upanishad's [[Narada Sanatkumara Samvada (नारदसनत्कुमारयोः संवादः)|Narada Sanathkumara Samvada]], clearly elucidates the existence of the subjects of sciences and arts depicting their antiquity in ancient India. Narada, entreated by Sanathkumara, enumerates the various sciences and arts studied by him and this list includes astronomy (nakshatra-vidya) and arithmetic (rasi-vidya). The culture of science of astronomy and mathematics classified under Aparavidya were not considered to be a hindrance to Paravidya or spiritual knowledge; they were part of the Chaturdasha and Asthadasa vidyas which was the basic curriculum of education. On the contrary they were considered as helpful adjuncts and were studied to aid the progress of Paravidya as expounded in Mundakopanishad (1.1.3-5).
== Contribution of Ancient and Medieval Indian Mathematicians ==
== Contribution of Ancient and Medieval Indian Mathematicians ==
Briefly the Dr. A. K. Bag discusses the contribution of Indian mathematicians to the world of computation, calculation and mathematics as follows.<ref>Bag, A. K., (1979) ''Mathematics in Ancient and Medieval India.'' Varanasi: Chaukhambha Orientalia. </ref>
Briefly the Dr. A. K. Bag discusses the contribution of Indian mathematicians to the world of computation, calculation and mathematics as follows.<ref name=":2">Bag, A. K., (1979) ''Mathematics in Ancient and Medieval India.'' Varanasi: Chaukhambha Orientalia. </ref>
* Expression of very large numbers by means of indices to ten and their use of fractions of various types.
* Expression of very large numbers by means of indices to ten and their use of fractions of various types.
* Concept of nine numerals, decimal place-value, with the introduction of zero significantly contributed to the development of Arithmetic.
* Concept of nine numerals, decimal place-value, with the introduction of zero significantly contributed to the development of Arithmetic.
* Development of the concept of infinitesimal (integral) calculus occurred in Yuktibhasa in connection with the summation of Infinite Series prior to western discoveries.
* Development of the concept of infinitesimal (integral) calculus occurred in Yuktibhasa in connection with the summation of Infinite Series prior to western discoveries.
== Sources on Mathematics ==
== Sources of Ancient Mathematics ==
=== Vedic Sources ===
=== Vedic Sources ===
Critical mathematical knowledge of Sutra charana literature led us to assume that there are mathematical works of even earlier age but they are lost. Shulbasutra-karas, seven of them, namely, Baudhayana, Apastamba, Katyayana, Manava, Maitrayana, Varaha, and Hiranyakeshi are regarded as the geometricians of the Vedic times. The Shulba works of these authors give methods and solutions of various problems of construction of altars required for yajnas.
Critical mathematical knowledge of Sutra charana literature led us to assume that there are mathematical works of even earlier age but they are lost. Shulbasutra-karas, seven of them, namely, Baudhayana, Apastamba, Katyayana, Manava, Maitrayana, Varaha, and Hiranyakeshi are regarded as the geometricians of the Vedic times. The Shulba works of these authors give methods and solutions of various problems of construction of altars required for yajnas.
Baudhayana shulbasutras are one of the earliest texts that describe the problems of construction of yajna-kundas. These texts also gave a general enunciation of the Pythogoras theorem, an approximate value of square-root of two correct to five places of decimal,
Baudhayana shulbasutras are one of the earliest texts that describe the problems of construction of yajna-kundas. These texts also gave a general enunciation of the Pythagoras theorem, an approximate value of square-root of two correct to five places of decimal, construction of a square equal to sum or difference of two squares, various methods of transformation of one figure to another etc.<ref name=":2" />
=== Jaina Works ===
Amongst the Jaina works the most important from the mathematical point of view include Suryaprajnapti, Sthanangasutra, Bhagavatisutra, Tattvarthadhigamasutra of Umasvati, Anuyogadvarasutra, Kshetrasamasa, Trilokasara etc. There were other mathematical works of the Jainas which are said to be lost.
=== Siddhanta Works ===
Eighteen siddhantas were composed during this period.
== Commentaries in Indian Mathematics ==
While there have been several extensive investigations on the history and achievements of the Indian tradition of sciences, there has not been much discussion on the foundational methodology of Indian sciences. Traditionally, such issues have been dealt with in the detailed bhashyas or commentaries, which continued to be written till recent times and played a vital role in the traditional scheme of learning.
As regards Indian mathematics, it is in such commentaries that we find detailed upapattis or "proofs" of the results and procedures, apart from a discussion of methodological and philosophical issues. It has been the scant attention paid, by the modern scholarship of the last two centuries, to this extensive tradition of commentaries, which has led to a lack of comprehension of the methodology of Indian mathematics.<ref name=":1" />
In his Vāsanābhāshya on his own treatise on Algebra, Bījaganita, Bhāskarācārya II (c.1150) explains that the tradition of upapatti has been for long a part of the oral instruction (pāñha-nibaddhā).3 The following are some of the important commentaries which are available in print and contain some discussion of upapattis for various results and procedures of Indian mathematics and astronomy:
1. Bhashya of Bhāskara I (c.629) on Āryabhañīya of Āryabhaña (c.499) 2. Bhāùya of Govindasvāmin (c.800) on Mahābhāskarīya of Bhāskara I (c.629) 3. Vāsanābhāùya of Caturveda Pçthūdakasvāmin (c.860) on Brāhmasphuñasiddhānta of Brahmagupta (c.628) 4. Vivaraõa of Bhāskarācārya II (c.1150) on Śiùyadhīvçddhidatantra of Lalla (c.748), 5. Vāsanā of Bhāskarācārya II (c.1150) on his own Līlāvatī, Bījagaõita and Siddhāntaśiromaõi 6. Siddhāntadīpikā of Parameśvara (c.1431) on the Bhāùya of Govindasvāmin (c.800) on Mahābhāskarīya of Bhāskara I (c.629) 7. Āryabhañīyabhāùya of Nīlakaõñha Somayājī (c.1501) on Āryabhañīya of Āryabhaña (c.499), K. Sambasiva Sastri (ed.), 3 Vols., Trivandrum 1931, 1932, 1957 8. Gaõita-Yuktibhāùā (in Malayalam) of Jyeùñhadeva (c.1530) 9. Yuktidīpikā of Śaïkara Vāriyār (c.1530) on Tantrasaïgraha of Nīlakaõñha Somayājī (c.1500) 10.Kriyākramakarī of Śaïkara Vāriyār (c.1535) on Līlāvatī of Bhāskarācārya II (c.1150) 11.Sūryaprakāśa of Sūryadāsa (c.1538) on Bījagaõita of Bhāskarācārya II (c.1150) 12.Buddhivilāsinī of Gaõeśa Daivajña (c.1545) on Līlāvatī of Bhāskarācārya II (c.1150)
== References ==
== References ==
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[[Category:Shastras]]
[[Category:Shastras]]