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History of Hindu Mathematics: A Source Book v1.c8.s1: 1. General Survey

History of Hindu Mathematics: A Source Book v1.c8.s1

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English

1. General Survey

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**Terminology and Scope.** Arithmetic forms the major part of the Hindu works on *pâṭîgaṇita*. The word *pâṭîgaṇita* is a compound formed from the words *pâṭî*, meaning "board," and *gaṇita*, meaning "science of calculation;" hence it means the science of calculation which requires the use of writing material (the board).¹ It is believed that this term originated in a non-Sanskrit literature of India, a vernacular of Northern India. The oldest Sanskrit term for the board is *phalaka* or *paṭṭa*, not *pâṭî*. The word *pâṭî* seems to have entered into Sanskrit literature about the beginning of the seventh century A.D.² The carrying out of mathematical calculations was sometimes called *dhûlî-karma* ("dust-work"), because the figures were written on dust spread on a board or on the ground. Some later writers have used the term *vyakta-gaṇita* ("the science of calculation by the 'known'") for *pâṭîgaṇita* to distinguish it from algebra which was called *avyakta-gaṇita* ("the science of calculation by the 'unknown'"). The terms *pâṭîgaṇita* and *dhûlî-karma* were translated into Arabic when Sanskrit works were rendered into that language. The Arabic equivalents are *ilm-hisâb-al-takht* ("the science of calculation on the board") and *hisâb-al-ghobâr* ("calculation on dust") respectively.

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¹ Paper being scarce, a wooden board was generally used for making calculations even upto the 19th century.

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² B. Datta, *American Math. Monthly*, XXXV, p. 526.

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Bayley, Fleet and several others suspect that the origin of the term *pâtî* in Hindu Mathematics lies in the use of the board as an abacus. This conjecture, however, is without foundation, as no trace of the use of any form of the abacus is found in India.

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According to Brahmagupta¹ there are twenty operations and eight determinations in *pâtîganita*. He says:

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¹ *BrSpSi*, p. 172.

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"He who distinctly and severally knows the twenty logistics, addition, etc., and the eight determinations including (measurement by) shadow is a mathematician."

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The twenty logistics, according to Pṛthudakasvâmî, are: (1) *saṁkalita* (addition), (2) *vyavakalita* or *vyutkalita* (subtraction), (3) *guṇana* (multiplication), (4) *bhâgahâra* (division), (5) *varga* (square), (6) *varga-mûla* (square-root), (7) *ghana* (cube), (8) *ghana-mûla* (cube-root), (9-13) *pañca jâti* (the five rules of reduction relating to the five standard forms of fractions), (14) *trairâśika* (the rule of three), (15) *vyasta-trairâśika* (the inverse rule of three), (16) *pañcarâśika* (the rule of five), (17) *sapta-râśika* (the rule of seven), (18) *navarâśika* (the rule of nine), (19) *ekâdaśarâśika* (the rule of eleven), and (20) *bhâṇḍa-pratibhâṇḍa* (barter and exchange). The eight determinations are: (1) *miśraka* (mixture), (2) *średhî* (progression or series), (3) *kṣetra* (plane figures), (4) *khâta* (excavation), (5) *citi* (stock), (6) *krâkacika* (saw), (7) *râśi* (mound), and (8) *châyâ* (shadow).

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Of the operations named above, the first eight have been considered to be fundamental by Mahâvîra and later writers. The operations of duplation (doubling)

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and mediation (halving), which were considered fundamental by the Egyptians, the Greeks and some Arab and western scholars, do not occur in the Hindu mathematical treatises. These operations were essential for those who did not know the place-value system of notation. They are not found in Hindu works, all of which use the place-value notation.

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**Sources.** The only works available which deal exclusively with *pâtîgaṇita* are: the *Bakhshâlî Manuscript* (*c.* 200), the *Triśatikâ* (*c.* 750), the *Gaṇita-sâra-saṁgraha* (*c.* 850), the *Gaṇita-tilaka* (1039), the *Lîlâvatî* (1150), the *Gaṇita-kaumudî* (1356), and the *Pâṭî-sâra* (1658). These works contain the twenty operations and the eight determinations mentioned above. Examples are also given to illustrate the use of the rules enunciated.

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Besides these there are a number of astronomical works, known as *Siddhânta*, each of which contains a section dealing with mathematics. Âryabhaṭa I (499) was the first to include a section on mathematics in his *Siddhânta*, the *Âryabhaṭîya*. Brahmagupta (628) followed Âryabhaṭa in this respect, and after him it became the general fashion to include a section on mathematics in a *Siddhânta* work.¹ The earlier *Siddhânta* works do not possess this feature. The *Sûrya-siddhânta* (*c.* 300) does not contain a section on mathematics. The same is true of the *Vâsiṣṭha*, the *Pitâmaha* and the *Romaka* Siddhântas. Bhâskara I and Lalla,² although zealous followers of Âryabhaṭa I, did not emulate him in including a section on mathematics in their astronomical works.

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¹ Amongst such works may be mentioned the *Mahâ-siddhânta* (950), the *Siddhânta-śekhara* (1036), the *Siddhânta-tattva-viveka* (1658), etc.

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² It is stated by Bhâskara II that Lalla wrote a separate treatise on *pâṭîgaṇita*.

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**Exposition and Teaching.** In India conciseness of composition, especially in scientific matters, was highly prized. The more compact and brief the composition, the greater was its value in the eyes of the learned. It is for this reason that the Indian treatises contain only a brief statement of the known formulæ and results, sometimes so concisely expressed as to be hardly understandable. This compactness is more pronounced in the older works; for instance, the exposition in the *Āryabhaṭîya* is more compact than in the later works.

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This hankering after brevity, in early times, was due chiefly to the dearth of writing material, the fashion of the time and the method of instruction followed. The young student who wanted to learn *pâṭîgaṇita* was first made to commit to memory all the rules. Then he was made to apply the rules to the solution of problems (also committing the problems to memory). The calculations were made on a *pâṭî* on which dust was spread, the numbers being written on the dust with the tip of the fore-finger or by a wooden style, the figures not required being rubbed out as the calculation proceeded. Sometimes a piece of chalk or soap-stone was used to write on the *pâṭî*. Along with each step in the process of calculation the *sûtra* (rule) was repeated by the student, the teacher supervising and helping the student where he made mistakes. After the student had acquired sufficient proficiency in solving the problems contained in the text he was studying, the teacher set him other problems—a store of graded examples (probably constructed by himself or borrowed from other sources) being the stock-in-trade of every professional teacher. At this stage the student began to understand and appreciate the *rationale* of the easier rules. After this stage was reached the teacher gave proofs of the more difficult formulæ to the pupil.

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It will be observed that the method of teaching pursued was extremely defective in so far as it was in the first two stages purely mechanical. A student who did not complete all the three stages knew practically nothing more than the mere mechanical application of a set of formulæ committed to memory; and as he did not know the *rationale* of the formulæ he was using, he was bound to commit mistakes in their application. It may be mentioned that not many teachers themselves could guide a pupil through all the stages of the teaching, and the earnest student, if he had a genuine desire to learn, had to go to some seat of learning or to some celeberated scholar to complete his training.

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Mathematics is and has always been the most difficult subject to study, and as a knowledge of higher mathematics could not be turned to material gain there were very few who seriously undertook its study. In India, however, the religious practices of the Hindus required a certain amount of knowledge of astronomy and mathematics. Moreover, there have always been, from very early times, a class of people known as *gaṇaka* whose profession was fortune-telling. These people were astrologers, and in order to impress their clients with their learning, they used to have some knowledge of mathematics and astronomy. Thus it would appear that instruction in mathematics, upto a certain minimum standard, was available almost everywhere in India. As always happens, some of the pupils got interested in mathematics for its own sake, and took pains to make a thorough study of the subject and to add to it by writing commentaries or independent treatises.

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**Decay of Mathematics.** All this was true when the times were normal. In abnormal times when there were foreign invasions, internal warfares or bad government and consequent insecurity, the study of mathematics and, in fact, of all sciences and arts languished. Al-Bîrûnî who visited north-western India after it had been in a very unsettled state due to recurrent Afghan invasions for the sake of plunder and loot complains that he could not find a *paṇḍit* who would explain to him the principles of Indian mathematics. Although Al-Bîrûnî's case was peculiar, for no respectable *paṇḍit* would agree to help a foreigner, especially one belonging to the same class as the invaders and the despoilers of temples, yet we are quite sure that in the Punjab there were very few good scholars at that time. We, however, know of at least one very distinguished mathematician, Śripati, who probably lived in Kashmir at that time.

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It is certain, however, that after the 12th century very little original work was done in India. Commentaries on older works were written and some new works brought out, but none of these had sufficient merit as regards exposition or subject matter, so as to displace the works of Bhâskara II, which have held undisputed sway for nine centuries (as standard text books).

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**The Fundamental Operations.** The eight fundamental operations of Hindu *gaṇita* are: (1) addition, (2) subtraction, (3) multiplication, (4) division, (5) square, (6) square-root, (7) cube and (8) cube-root. Most of these elementary processes have not been mentioned in the Siddhânta works. Âryabhaṭa I gives the rules for finding the square- and cube-roots only, whilst Brahmagupta gives the cube-root rule only. In the works on arithmetic (*pâṭîgaṇita*), the methods of addition and subtraction have not been mentioned at all or mentioned very briefly. Names of several methods of multiplication have been mentioned, but the methods themselves have been either very briefly described or not described at all. The modern method of division is briefly described in all the works and so are the methods of squaring, square-root, cubing and cube-root.

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Although very brief descriptions of these fundamental operations are available, yet it is not difficult to reconstruct the actual procedure employed in performing these operations in ancient India. These methods have been well-known and taught to children, practically without any change, for the last fifteen hundred years or more. They are still performed in the old fashion on a *pâṭî* ("board") by those who have obtained their primary training in the Sanskrit pâṭhaśâlâ and not in the modern primary school. The details of these methods are also available to us in the various commentaries, viz., the commentary of Pṛthudakasvâmî and the several commentaries on Bhâskara's *Lîlâvatî*.

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As already mentioned, the calculations were performed on sand spread on the ground (*dhûlî-karma*¹) or on a *pâṭî* ("board"). Sometimes a piece of chalk or soap-stone (*pâṇḍu-lekha* or *śvetavarṇî*) was used to write on the *pâṭî*.² As the figures written were big, so several lines of figures could not be contained on the board. Consequently, the practice of obliterating figures not required for subsequent work was common. Instances of this would be found in the detailed method of working (the operations) given hereafter.

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¹ Bhâskara II, *SiŚi*, *candragrahaṇâdhikâra*, 4.

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² Bhâskara II: *khaṭikâyâ rekhâ ucchâdya*..., *i.e.*, "having drawn lines with a chalk...," quoted by S. Dvivedi in his *History of Mathematics* (in Hindi), Benares, 1910, p. 41.

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That all mathematical operations are variations of the two fundamental operations of addition and subtraction was recognised by the Hindu mathematicians from early times. Bhâskara I (*c.* 525) states:¹

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¹ The quotation is from his commentary on the *Âryabhaṭîya*.

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“All arithmetical operations resolve into two categories though usually considered to be four.² The two main categories are increase and decrease. Addition is increase and subtraction is decrease. These two varieties of operations permeate the whole of mathematics (*gaṇita*). So previous teachers have said: ‘Multiplication and evolution are particular kinds of addition; and division and involution of subtraction. Indeed every mathematical operation will be recognised to consist of increase and decrease.’ Hence the whole of this science should be known as consisting truly of these two only.”

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² *i.e.*, addition, subtraction, multiplication and division.

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