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 । तद्वद्वेदाङ्गशास्राणां 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 Indian mathematicians to the world of computation, calculation and mathematics as follows.

• 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.
• 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 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.

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)