Ganita Shastra (गणितशास्त्रम्)

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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.

यथा शिखा मयूराणां नागानां मणयो यथा । तद्वद्वेदाङ्गशास्त्राणां ज्योतिषं (गणितं) मूर्धनि स्थितम् ॥ १९ ॥ (Veda. Jyot. 4)

yathā śikhā mayūrāṇāṁ nāgānāṁ maṇayo yathā । tadvadvedāṅgaśāsrāṇāṁ jyotiṣaṁ (gaṇitaṁ) mūrdhani sthitam ॥ 19 ॥

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 or the six ancillary branches of knowledge).

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), 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.

Chandogya Upanishad's 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).

The elementary stage in ancient education system lasted from the age of five till about the age of twelve. The main subjects of study were Lipi (लिपि) or lekha (alphabets, reading and writing), rupa (drawing and geometry) and ganita (arithmetic). It is said in the Arthashastra of Kautilya that having undergone the Choula ceremony (tonsure), the student shall learn the alphabets (lipi) and arithmetic (संख्यानम् । samkhyana).

वृत्त-चौल-कर्मा लिपिं संख्यानं चौपयुञ्जीत ।। ०१.५.०७ ।। (Arth. Shast. 1.5.7)

Mention of lekha, rupa and ganana is also found in the Jaina canonical works.

Our ancestors were well aware that mathematics is the language by which one can express any associated science compactly. It involved simplified ways of manipulation and calculations, helping one to develop the mind and also to communicate well. Thus mathematics was considered to be a very important subject in ancient India taught from a very young age.

Scope of Ganita

In Vedangas Astronomy (jyotisha) became a separate subject and geometry (kshetra-ganita) came to be included within its scope. It is to be noted that the Jain and Buddhist scholars have played a significant role in the development of astronomy and mathematics in India. Importance of Ganita is also given by the Jainas in their religious literature. Ganitanuyoga, meaning the exposition of the principles of mathematics and Samkhyana meaning the science of numbers, in terms of arithmetic and astronomy, is stated to be one of the principal accomplishments of a Jaina priest (Bhagavati-sutra, 90 and Uttaradhyayana-sutra, 25). Buddhist literature regards arithmetic (ganana and samkhyana) as the first and noblest of the arts. In ancient Buddhist literature we find a mention of three classes of Ganita:

1. Mudra ("finger arithmetic")
2. Ganana ("mental arithmetic")
3. Samkhyana ("higher arithmetic in general")

Subject Matter of Ganita

The subjects treated in Ganita, in early centuries, consisted of the following:

• Parikarma ("fundamental operations")
• Vyavahara ("determinations")
• Rajju ("rope," meaning geometry)
• Rasi ("rule of three")
• Kalasavarna ("operations with fractions")
• Yavat tavat ("as many as," meaning simple equations)
• Varga ("Square," meaning quadratic equations)
• Ghana ("Cube", meaning cubic equations)
• Varga-varga (biquadratic equations)
• Vikalpa ("permutations and combinations")

Thus ganita came to mean mathematics in general, while 'finger arithmetic' as well as 'mental arithmetic' were excluded from the scope of its meaning. For the calculations involved in ganita, the use of some writing material became essential. The calculations were performed on a board (patti) with a piece of chalk or on sand (dhuli) spread on the ground or on the patti, Thus the terms patti-ganita ("science of calculation on the board") or dhuli-karma ("dust-work"), came to be used for higher mathematics. Later on the section of ganita dealing with algebra was given the name Bija-ganita. The first to effect this separation was Brahmagupta, but he did not use the term Bijaganita. The chapter dealing with algebra in his Brahma-sphuta-siddhanta is called Kuttaka. Sridharacarya (750) regarded Pati-ganita and Bijaganita as separate and wrote separate treatises on each. This distinction between Pati-ganita and Bijaganita has been preserved by later writers.

The development of numerical symbolism is intimately connected with the writing in ancient India.

Contribution of Ancient and Medieval Indian Mathematicians

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.

• 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.
• Formulation of rule of three methods of calculation
• Accurate calculation of Surd numbers.
• Geometrical knowledge and classification of geometrical figures on the basis of angles and sides was provided as early in as the Sulbasutras
• Pythogoras theorem was elucidated
• Calculation of the value of squareroot of 2 by geometrical method.
• Determination of diagonals of cyclic quadrilateral and methods for the construction of a quadrilateral given by Brahmagupta
• In the field of Algebra - symbols of operation, equation, rule of false position, quadratic equation, indeterminate equations of first and second degree, progressive series, permutations and combination, and Pascal's triangle with binomial theorem have been discussed.
• Development of trigonometrical formulae, sine table construction, and value of pi.
• Work on trigonometrical series - pi, sine, cosine, and tan was first carried out by the Indians more than a century before Newton (1664 CE) and Leibniz (1676 CE).
• Development of the concept of infinitesimal (integral) calculus occurred in Yuktibhasa in connection with the summation of Infinite Series prior to western discoveries.

Sources of Ancient Mathematics

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.

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.

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 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.

• Surya Siddhanta
• Paitamaha Siddhanta
• Vyasa Siddhanta
• Vashishta Siddhanta
• Atri Siddhanta
• Parasara Siddhanta
• Kashyapa Siddhanta
• Narada Siddhanta
• Garga Siddhanta
• Marichi Siddhanta
• Manu Siddhanta
• Angira Siddhanta
• Lomasa Siddhanta
• Paulisa Siddhanta
• Chyavana Siddhanta
• Yavana Siddhanta
• Bhrugu Siddhanta
• Saunaka Siddhanta

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).

Modern Indian Mathematicians

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.

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. Govindasvamin (800-850 A.D.) - Bhashya on Mahabhaskariya of Bhaskara I and Govindakrti. He belonged to the Kerala school of mathematics.
7. Mahavira (850 A.D.) - Ganitasarasamgraha deals with arithmetic, geometry and algebra. A Jaina mathematician he is associated with the school of Mysore.
8. 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.
9. 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.
10. 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.
11. 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.

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.

1. Narayana Pandita (1356 A.D.) - Ganitakaumudi, a mathematical treatise and Bijaganitavatamsa, a work on algebra. Known as Narayana Daivajna, he was solely a mathematician.
2. 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).
3. 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.
4. 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.
5. Jnanaraja (1503 A.D.) - Siddhantasundarabija and Siddhatasundara, both of which remain unpublished.
6. Chitrabhanu (1475 - 1550 A.D.) - Karnaamrta, is a work on mathematics.

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.

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:

• Bhashya of Bhāskara I (c.629) on Āryabhañīya of Āryabhaña (c.499)
• Bhāùya of Govindasvāmin (c.800) on Mahābhāskarīya of Bhāskara I (c.629) Vāsanābhāùya of Caturveda Pçthūdakasvāmin (c.860) on Brāhmasphuñasiddhānta of Brahmagupta (c.628)
• Vivaraõa of Bhāskarācārya II (c.1150) on Śiùyadhīvçddhidatantra of Lalla (c.748),
• Vāsanā of Bhāskarācārya II (c.1150) on his own Līlāvatī, Bījagaõita and Siddhāntaśiromaõi
• Siddhāntadīpikā of Parameśvara (c.1431) on the Bhāùya of Govindasvāmin (c.800) on Mahābhāskarīya of Bhāskara I (c.629)
• Āryabhañīyabhāùya of Nīlakaõñha Somayājī (c.1501) on Āryabhañīya of Āryabhaña (c.499), K. Sambasiva Sastri (ed.), 3 Vols., Trivandrum 1931, 1932, 1957
• Gaõita-Yuktibhāùā (in Malayalam) of Jyeùñhadeva (c.1530)
• Yuktidīpikā of Śaïkara Vāriyār (c.1530) on Tantrasaïgraha of Nīlakaõñha Somayājī (c.1500)
• Kriyākramakarī of Śaïkara Vāriyār (c.1535) on Līlāvatī of Bhāskarācārya II (c.1150)
• Sūryaprakāśa of Sūryadāsa (c.1538) on Bījagaõita of Bhāskarācārya II (c.1150)
• Buddhivilāsinī of Gaõeśa Daivajña (c.1545) on Līlāvatī of Bhāskarācārya II (c.1150)