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Ganita Shastra (गणितशास्त्रम्) (view source)

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Ganita shastra (Samskrit: गणितशास्त्रम्) or Ganita (गणितम्) means the science of calculation which is an equivalent name for the term mathematics. This ancient term occurs copiously in Vedic literature. Vedanga Jyotisha by Lagadha gives it the highest place of honour among the sciences which form the Vedanga. <blockquote>यथा शिखा मयूराणां नागानां मणयो यथा । तद्वद्वेदाङ्गशास्राणां ज्योतिषं (गणितं) मूर्धनि स्थितम् ॥ १९ ॥ (Veda. Jyot. 4)<ref name=":0">B.B. Datta and A. N. Singh (1962) ''History of Hindu Mathematics, A Source Book, Parts 1 and 2.'' Bombay: Asia Publishing House. (Page 7)</ref></blockquote><blockquote>yathā śikhā mayūrāṇāṁ nāgānāṁ maṇayo yathā । tadvadvedāṅgaśāsrāṇāṁ jyotiṣaṁ (gaṇitaṁ) mūrdhani sthitam ॥ 19 ॥</blockquote>As the crests on the heads of peacocks, as the gems on the hoods of the snakes (cobras), so is astronomy (mathematics) is at the highest position of vedanga shastras (which are the [[Shad Vedangas (षड्वेदाङ्गानि)|Shad Vedangas]] or the six ancillary branches of knowledge).<ref name=":0" />
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Ganita shastra (Samskrit: गणितशास्त्रम्) or Ganita (गणितम्) means the science of calculation which is an equivalent name for the term mathematics. This ancient term occurs copiously in Vedic literature. Vedanga Jyotisha by Lagadha gives it the highest place of honour among the sciences which form the Vedanga. <blockquote>यथा शिखा मयूराणां नागानां मणयो यथा । तद्वद्वेदाङ्गशास्त्राणां ज्योतिषं (गणितं) मूर्धनि स्थितम् ॥ १९ ॥ (Veda. Jyot. 4)<ref name=":0">B.B. Datta and A. N. Singh (1962) ''History of Hindu Mathematics, A Source Book, Parts 1 and 2.'' Bombay: Asia Publishing House. (Page 7)</ref></blockquote><blockquote>yathā śikhā mayūrāṇāṁ nāgānāṁ maṇayo yathā । tadvadvedāṅgaśāsrāṇāṁ jyotiṣaṁ (gaṇitaṁ) mūrdhani sthitam ॥ 19 ॥</blockquote>As the crests on the heads of peacocks, as the gems on the hoods of the snakes (cobras), so is astronomy (mathematics) is at the highest position of vedanga shastras (which are the [[Shad Vedangas (षड्वेदाङ्गानि)|Shad Vedangas]] or the six ancillary branches of knowledge).<ref name=":0" />
    
== Introduction ==
 
== Introduction ==
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== Contribution of Ancient and Medieval Indian Mathematicians ==
 
== Contribution of Ancient and Medieval Indian Mathematicians ==
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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>
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Briefly the Dr. A. K. Bag discusses the contribution of ancient and modern 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>
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* Expression of very large numbers by means of indices to ten and their use of fractions of various types.
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* Numeration in India was known since the earliest times. Expression of very large numbers by means of indices to ten and their use of fractions of various types has been mentioned in the Samhitas.
 
* 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.
 
* Formulation of rule of three methods of calculation
 
* Formulation of rule of three methods of calculation
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=== Siddhanta Works ===
 
=== Siddhanta Works ===
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Eighteen siddhantas were composed during this period.  
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Eighteen siddhantas were composed by and named after their authors. However five of them primarily, Surya, Paulisa, Romaka, Vashishta, Paitamaha siddhantas and a few others have survived, while the remaining are lost. Scholars also opine that these works have undergone various changes over time. 
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{{columns-list|colwidth=15em|style=width: 600px; font-style: italic;|
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* [[Surya Siddhanta]]
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* [[Paitamaha Siddhanta]]
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* [[Vyasa Siddhanta]]
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* [[Vashishta Siddhanta]]
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* [[Atri Siddhanta]]
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* [[Parasara Siddhanta]]
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* [[Kashyapa Siddhanta]]
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* [[Narada Siddhanta]]
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* [[Garga Siddhanta]]
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* [[Marichi Siddhanta]]
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* [[Manu Siddhanta]]
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* [[Angira Siddhanta]]
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* [[Lomasa Siddhanta]]
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* [[Paulisa Siddhanta]]
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* [[Chyavana Siddhanta]]
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* [[Yavana Siddhanta]]
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* [[Bhrugu Siddhanta]]
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* [[Saunaka Siddhanta]]
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}}
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Astronomy which had been a branch of Mathematics, separated out and began to be regarded by the name Jyotisha. Geometry which belonged to the Kalpasutras came to be regarded as an integral part of the ganita. Thus the readjusted science of the ganita consisted mainly of arithmetic, algebra, and geometry. Among the Jainas and Buddhists the ganita was known by the name Samkyana (science of numbers).
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== Modern Indian Mathematicians ==
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Three schools - Ujjain, Mysore and Kusumpura have been referred to as important schools of mathematics in the early christian era. Here we discuss the important mathematicians of the modern era and their significant contributions to astronomy and mathematics.
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# '''Aryabhata I (476 A.D.)''' - Aryabhatiya or Aryasiddhanta consists of four chapters namely Dasagitika (the ten Gitikas), Ganitapada (mathematics), Kalakriya (reckoning of time) and Gola (sphere) deals with astronomy and arithmetic.
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# '''Varahamihira (505 A.D.)''' - Panchasiddhantika among other works is considered important in the history of astronomy. In the history of mathematics this work has a high place for its amount of trigonometrical information.
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# '''Bhaskara I (600 A.D.)''' - Mahabhaskariya, an astronomical work, Aryabhatiyabhashya, a commentary on the Aryabhatiya of Aryabhata and Laghubhaskariya, an abridged and simplified version of Mahabhaskariya. He was mainly an astronomer but he propounded mathematical solutions for astronomical problems. He gave the solution of the indeterminate equation of the first degree in his Mahabhaskariya.
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# '''Brahmagupta (628 A.D.)''' - Brahmasphutasiddanta and Khandakhadyaka are astronomical works which include his greatest contributions to the field of mathematics. He belonged to the Ujjain school. Brahmasphutasiddhanta, has two chapters Ganitadhyaya and Kuttakadhyaya. Ganitadhyaya deals with cyclic triange and quadrilateral, rules for arithmetical operations involving zero, negative numbers and quadratic equations. Kuttakadhyaya contains solutions of the indeterminate equations of both first and second degree. The Spastadhikara chapter however contains trigonometrical notations including the sine table. He has a significant place in the history of Indian Civilization.
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# '''Lalla (768 A.D.)''' - Shishyadhivrddhida based on the Aryabhatiya of Aryabhata I, is devoted to astronomy. Associated with the school of Kusumpura his work contains important information on trigonometry.
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# '''Govindasvamin (800-850 A.D.)''' - Bhashya on Mahabhaskariya of Bhaskara I and Govindakrti. He belonged to the Kerala school of mathematics.
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# '''Mahavira (850 A.D.)''' - Ganitasarasamgraha deals with arithmetic, geometry and algebra. A Jaina mathematician he is associated with the school of Mysore.
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# '''Sridhara (850-950 A.D.)''' - Patiganita is a work on arithmetic and mensuration. Famous as Sridharacharya, he dealt with multiplication, division, square, cube, squareroot, cube-root, fraction, rule of three and the areas of plane figures. For the first time gave a rule to extract the root of ax²+bx = c, which is known usually as Sridhara's formula.
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# '''Aryabhata II (950 A.D.)''' - Mahabhaskariya, an astronomical work dealing with various problems of mathematical interest besides preliminary operations. He mentions separately about the three branches of mathematics, Pati, Kuttaka and Bija in this work.
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# '''Sripati (1039 A.D.)''' - Ganitatilaka, Siddhantasekhara and Bijaganita besides five other works on astronomy and astrology. A Jaina astronomer, his Ganitatilaka is devoted exclusively to arithmetic. The Siddhantasekhara, an astronomical work, however, deals with algebra in two chapters. Bijaganita is now lost.
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# '''Bhaskara II (1114 - 1200 A.D.)''' - Lilavati, Bijaganita and Siddhantasiromani are the famous works of Bhaskaracharya. He authored two other works Vasanabhaasya, his own commentary of the Siddhantasiromani, and Karanakutuhala, a treatise on planetary motion. The Lilavati which is based on Brahmagupta's Brahmasphutasiddhanta, Sridhara's Patiganita, and Mahasiddhanta of Aryabhata II exhibits a profound system of arithmetic and also contains many useful propositions in geometry and arithmetic. The full solution of the equation and of its more general form ax²+bx+c = y² was given by Bhaskaracharya II. He was acquainted with the principle of infinitesimal calculus and is often given credit for originating the idea of integration long before Newton and Leibniz.
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# 
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The Kerala school of mathematics developed from fourteenth century onwards. Kerala has been seat of learning and has produced great mathematical works building an unbroken tradition for about five hundred years. The mathematicians of Kerala were staunch followers of Aryabhata I. The activities of the scholars of this period who attained an independent distinction are still unpublished and are yet to be studied.<ref name=":2" />
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# '''Narayana Pandita (1356 A.D.)''' - Ganitakaumudi, a mathematical treatise and Bijaganitavatamsa, a work on algebra. Known as Narayana Daivajna, he was solely a mathematician. 
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# '''Madhava (1400 A.D.)''' - Venvaroha, a well known astronomical treatise. He is regarded as the authority on spherical astronomy and mathematics and is often referred to by later writers as a golavid (expert on spherical mathematics).
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# '''Paramesvara (1430 A.D.)''' - author of many commentaries on all the popular classical works on astronomy and mathematics besides some original works on astronomy such as Drgganita and Goladipika. He was the founder of the drgganita system of astronomy in Kerala though mainly an astronomer was held with respect in the field of mathematics.
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# '''Nilakantha Somasutvan (1443 - 1543 A.D.)''' - Aryabhattiyabhashya and Tantrasamgraha contribute much to mathematics. He is the author of many other astronomical works namely Golasara, Chandrachhayaganita, Chandrachhayaganita-tika.
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# '''Jnanaraja (1503 A.D.)''' - Siddhantasundarabija and Siddhatasundara, both of which remain unpublished.
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# '''Chitrabhanu (1475 - 1550 A.D.)''' - Karnaamrta, is a work on mathematics.
    
== Commentaries in Indian Mathematics ==
 
== Commentaries in Indian Mathematics ==
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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:  
 
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:  
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* Bhashya of Bhāskara I (c.629) on Āryabhañīya of Āryabhaña (c.499)
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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)
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* Bhāùya of Govindasvāmin (c.800) on Mahābhāskarīya of Bhāskara I (c.629) Vāsanābhāùya of Caturveda Pçthūdakasvāmin (c.860) on Brāhmasphuñasiddhānta of Brahmagupta (c.628)  
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* Vivaraõa of Bhāskarācārya II (c.1150) on Śiùyadhīvçddhidatantra of Lalla (c.748),  
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* Vāsanā of Bhāskarācārya II (c.1150) on his own Līlāvatī, Bījagaõita and Siddhāntaśiromaõi  
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* Siddhāntadīpikā of Parameśvara (c.1431) on the Bhāùya of Govindasvāmin (c.800) on Mahābhāskarīya of Bhāskara I (c.629)  
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* Āryabhañīyabhāùya of Nīlakaõñha Somayājī (c.1501) on Āryabhañīya of Āryabhaña (c.499), K. Sambasiva Sastri (ed.), 3 Vols., Trivandrum 1931, 1932, 1957  
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* Gaõita-Yuktibhāùā (in Malayalam) of Jyeùñhadeva (c.1530)  
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* Yuktidīpikā of Śaïkara Vāriyār (c.1530) on Tantrasaïgraha of Nīlakaõñha Somayājī (c.1500)  
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* Kriyākramakarī of Śaïkara Vāriyār (c.1535) on Līlāvatī of Bhāskarācārya II (c.1150)  
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* Sūryaprakāśa of Sūryadāsa (c.1538) on Bījagaõita of Bhāskarācārya II (c.1150)  
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* Buddhivilāsinī of Gaõeśa Daivajña (c.1545) on Līlāvatī of Bhāskarācārya II (c.1150)
    
== References ==
 
== References ==
 
<references />
 
<references />
 
[[Category:Shastras]]
 
[[Category:Shastras]]
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