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History of Hindu Mathematics: A Source Book v1.c8: Chapter II. Arithmetic

History of Hindu Mathematics: A Source Book v1.c8

Language
English

Contents

Chapter II. Arithmetic

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.

2. Addition

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**Terminology.** Âryabhaṭa II (950) defines addition thus:

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“The making into one of several numbers is addition.”³

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³ *MSi*, p. 143.

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The Hindu name for addition is *saṁkalita* (made together). Other equivalent terms commonly used are *saṁkalana* (making together), *miśraṇa* (mixing), *sammelana* (mingling together), *prakṣepaṇa* (throwing together), *saṁyojana* (joining together), *ekîkaraṇa* (making into one), *yukti*, *yoga* (addition) and *abhyâsa*,⁴ etc. The word *saṁkalita* has been used by some writers in the general sense of the sum of a series.⁵

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⁴ This word has been used in the sense of addition in the *Śulba* only. It is used for multiplication in later works.

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⁵ *E.g.*, *Triś*, p. 2; *GSS*, p. 17.

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**The Operation.** In all mathematical and astronomical works, a knowledge of the process of addition is taken for granted. Very brief mention of it is made in some later works of elementary character. Thus Bhâskara II says in the *Lîlâvatî*:

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"Add the figures in the same places in the direct or the inverse order."¹

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¹ *L*, p. 2; direct (*krama*), *i.e.*, beginning from the units place; inverse (*utkrama*), *i.e.*, beginning from the last place on the left. The commentator Gaṅgâdhara says: *aṅkânâṃ vâmatogatiriti vitarkeṇa ekasthânâdi yojanam kramaḥ utkramastu antyasthânâdi yojanam*, *i.e.*, "According to the rule 'the numerals increase (in value) towards the left', the addition of units first is the direct method, the addition of figures in the last place first is the inverse method."

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**Direct Process.** In the direct process of addition referred to above, the numbers to be added are written down, one below the other, just as at present, and a line is drawn at the bottom below which the sum is written. At first the sum of the numbers standing in the units place is written down, thus giving the first figure of the sum. The numbers in the tens place are then added together and their sum is added to the figure in the tens place of the partial sum standing below the line and the result substituted in its place. Thus the figure in the tens place of the sum is obtained; and so on. An alternative method used was to write the biggest addend at the top, and to write the digits of the sum by rubbing out corresponding digits of this addend.²

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² Dvivedi, *History of Mathematics*, Benares, 1910, p. 60.

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**Inverse Process.** In the inverse process, the numbers standing in the last place (extreme left) are added together and the result is placed below this last place. The numbers in the next place are then added and so on. The numbers of the partial sum are corrected, if necessary, when the figures in the next vertical line are added. For instance, if 12 be the sum of the numbers in the last place, 12 is put below the bottom line, 2 being directly below the numbers added; then, if the sum of the numbers in the next place is 13 (say), 3 is placed below the figures added and 1 is carried to the left. Thus the figure 2 of the partial sum 12 is rubbed out and substituted by 3.¹

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¹ The *Manorañjana* explains the process of addition thus: *Example.* Add 2, 5, 32, 193, 18, 10 and 100.

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The Arabs used to separate the places by vertical lines, but this was not done by the Hindus.²

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² *Cf.* Taylor, *Lîlâwatî*, Bombay, 1816, Introduction, p. 14.

3. Subtraction

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**Terminology.** Āryabhaṭa II (950) gives the following definition of subtraction:

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"The taking out (of some number) from the *sarvadhana* (total) is subtraction; what remains is called *śeṣa* (remainder)."³

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³ *MSi*, p. 143.

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The terms *vyutkalita* (made apart), *vyutkalana* (making apart), *śodhana* (clearing), *pâtana* (causing to fall), *viyoga* (separation), etc., have been used for subtraction. The terms *śeṣa* (residue) and *antara* (difference) have been used for the remainder. The minuend has been called *sarvadhana* or *viyojya* and the subtrahend *viyojaka*.

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**The Operation.** Bhâskara II gives the method of subtraction thus:

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"Subtract the numbers according to their places in the direct or inverse order."⁴

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⁴ *L*, p. 2.

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| Sum of units | 2,5,2,3,8,0,0 | 20 |
| Sum of tens | 3,9,1,1,0 | 14 |
| Sum of hundreds | 1,0,0,1 | 2 |
| Sum of sums | | 360 |

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The horizontal process has been adopted by the commentator so that both the 'direct' and 'inverse' processes may be exhibited by a single illustration. It was never used in practice.

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**Direct Process.** Sûryadâsa¹ explains the process of subtraction with reference to the example. 1000 — 360 thus:

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¹ In his commentary on the *Lîlâvatî*.

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“Hence making the subtraction as directed, six cannot be subtracted from the zero standing in the tens place, so taking ten and subtracting six from it, the remainder (four) is placed above (six), and this ten is to be subtracted from the next place. For, as the places of unit, etc., are multiples of ten, so the figure of the subtrahend that cannot be subtracted from the corresponding figure of the minuend is subtracted from ten, the remainder is taken and this ten is deducted from the next place. In this way this ten is taken to the last place until it is exhausted with the last figure. In other words, numbers upto nine occupy one place, the differentiation of places begins from ten, so it is known ‘how many tens there are in a given number’ and, therefore, the number that cannot be subtracted from its own place is subtracted from the next ten, and the remainder taken.”

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The above refers to the direct process, in which subtraction begins from the units place.

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**Inverse Process.** The inverse process is similar, the only difference being that it begins from the last place of the minuend, and the previously obtained partial differences are corrected, if required. The process is suitable for working on a *pâṭî* (board) where figures can be easily rubbed out and corrected. This process seems to have been in general use in India, and was considered to be simpler than the direct process.²

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² According to Gaṅgâdhara, the inverse process of working is easier in the case of subtraction, and the direct in the case of addition.

4. Multiplication

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**Terminology.** The common Hindu name for multiplication is *guṇana*. This term appears to be the oldest as it occurs in Vedic literature. The terms *hanana*, *vadha*, *kṣaya*, etc., which mean "killing" or "destroying" have been also used for multiplication. These terms came into use after the invention of the new method of multiplication with the decimal place-value numerals; for in the new method the figures of the multiplicand were successively rubbed out (destroyed) and in their places were written the figures of the product.¹ Synonyms of *hanana* (killing) have been used by Âryabhaṭa I² (499), Brahmagupta (628), Śrîdhara (*c.* 750) and later writers. These terms appear also in the Bakhshâlî Manuscript.³

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¹ See the *kapâṭa-sandhi* method of multiplication, pp. 138ff.

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² *Â*, ii. 19, 26, etc.

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³ *BMs*, 65 *verso*.

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The term *abhyâsa* has been used both for addition and multiplication in the *Śulba* works (800 B.C.). This shows that at that early period, the process of multiplication was made to depend on that of repeated addition. The use of the word *parasparakṛtam* (making together) for multiplication in the Bakhshâlî Manuscript⁴ is evidently a relic of olden times. This ancient terminology proves that the definition of multiplication was "a process of addition resting on repetition of the multiplicand as many times as is the number of the multiplicator." This definition occurs in the commentary of the *Âryabhaṭîya* by Bhâskara I. The commentators of the *Lîlâvatî* give the same explanation of the method of multiplication.⁵

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⁴ *BMs*, 3 *verso*.

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⁵ Colebrooke, *Hindu Algebra*, p. 133.

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The multiplicator was termed *guṇya* and the multiplier *guṇaka* or *guṇakāra*. The product was called *guṇana-phala* (result of multiplication) or *pratyutpanna* (lit. "reproduced," hence in arithmetic "reproduced by multiplication"). The above terms occur in all known Hindu works.

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**Methods of Multiplication.** Āryabhaṭa I does not mention the common methods of multiplication, probably because they were too elementary and too well-known to be included in a Siddhānta work. Brahmagupta, however, in a supplement to the section on mathematics in his Siddhānta, gives the names of some methods with very brief descriptions of the processes:

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"The multiplicand repeated, as in *gomūtrikā*, as often as there are digits¹ in the multiplier, is severally multiplied by them and (the results) added (according to places); this gives the product. Or the multiplicand is repeated as many times as there are component parts² in the multiplier."³

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¹ *khaṇḍa*, translated as "integrant portions" by Colebrooke.

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² *bheda*, i.e., portions which added together make the whole, or aliquot parts which multiplied together make the entire quantity.

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³ *BrSpSi*, p. 209; Colebrooke, *l.c.*, p. 319.

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"The multiplicand is multiplied by the sum or the difference of the multiplier and an assumed quantity and, from the result the product of the assumed quantity and the multiplicand is subtracted or added."⁴

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⁴ *BrSpSi*, p. 209. Colebrooke (*l.c.*, p. 320) thinks that this is a method to obtain the true product when the multiplier has been taken to be too great or too small by mistake. This view is incorrect.

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Thus Brahmagupta mentions four methods: (1) *gomūtrikā*, (2) *khaṇḍa*, (3) *bheda* and (4) *iṣṭa*. The common and well-known method of *kapāṭa-sandhi* has been omitted by him.

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Śrîdhara mentions four methods of multiplication: (1) *kapâṭa-sandhi*, (2) *tastha*, (3) *rûpa-vibhâga* and (4) *sthâna-vibhâga*. Mahâvîra mentions the same four. Âryabhaṭa II mentions only the common method of *kapâṭa-sandhi*. Bhâskara II, besides the above four, mentions Brahmagupta's method of *iṣṭa-guṇana*. The five methods given by Bhâskara II were mentioned earlier by Śrîpati in the *Siddhânta-śekhara*. Gaṇeśa¹ (1545) mentions the gelosia method of multiplication under the name of *kapâṭa-sandhi* and adds that the intelligent can devise many more methods of multiplication. The method is also given in the *Gaṇita-mañjarî*. We have designated it as *kapâṭa-sandhi (b)*.

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¹ Commentary on the *Lîlâvatî*, MSS No. I. B. 6. in the Asiatic Soc. of Bengal, Calcutta, pp. 17, 18. In this work only two methods are given, (1) *kapâṭa-sandhi* and (2) *kapâṭa-sandhi (b)*.

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Seven² distinct modes of multiplication employed by the Hindus are given below. Some of these are as old as 200 A.D. These methods were transmitted to Arabia in the eighth century and were thence communicated to Europe, where they occur in the writings of mediæval mathematicians.

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² Or ten if we count also the sub-divisions under each head.

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**Door-junction Method.** The Sanskrit term for the method is *kapâṭa-sandhi*. Śrîdhara³ describes it thus:

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³ *Tris*, pp. 3f.

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"Placing the multiplicand below the multiplier as in *kapâṭa-sandhi*,⁴ multiply successively, in the direct or inverse order, moving the multiplier each time. This method is called *kapâṭa-sandhi*."

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⁴ *kapâṭa* means "door" and *sandhi* means "junction"; hence *kapâṭa-sandhi* means "the junction of doors."

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Âryabhaṭa II⁵ (950) gives the following without name:

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⁵ *MSi*, p. 143; the inverse method only has been given.

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"Place the first figure of the multiplier over the last figure of the multiplicand, and then multiply successively all the figures of the multiplier by each figure of the multiplicand."

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Śrîpati¹ (1039) gives the name *kapâṭa-sandhi* and states:

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¹ *SiŚe*, xiii. 2; *GT*, 15.

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"Placing the multiplicand below the multiplier as in the junction of two doors multiply successively (the figures of the multiplicand) by moving it (the multiplier) in the direct or inverse order."

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Mahâvîra refers to a method known as *kapâṭa-sandhi*, but does not give the details of the process.² Bhâskara II gives the method but not the name, while Nârâyaṇa (1356) gives the method in almost the same words as Śrîdhara, and calls it *kapâṭa-sandhi*.

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² *GSS*, p. 9.

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The main features of the method are (*i*) the relative positions of the multiplicand and the multiplier and (*ii*) the rubbing out of figures of the multiplicand and the substitution in their places of the figures of the product. The method owes its name *kapâṭa-sandhi* to the first feature, and the later Hindu terms meaning "killing" or "destroying" for multiplication owe their origin to the second feature. The occurrence of the terms *hanana*, *vadha*, etc., in the works of Âryabhaṭa I and Brahmagupta, and in the Bakhshâlî Manuscript show beyond doubt that this method was known in India about 200 A.D.

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The following illustrations³ explain the two processes of multiplication according to the *kapâṭa-sandhi* plan:

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³ The illustrations are based on the accounts given in the commentaries on the *Lîlâvatî*, especially the *Manorañjana* which gives more details.

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*Direct Process*: This method of working does not appear to have been popular. It has not been mentioned by writers after the 11th century, Śrîpati (1039) being the last writer to mention it.

v1.c8.s4.37

*Example*. To multiply 135 by 12.

v1.c8.s4.38

The numbers are written down on the *pâṭî* thus:

v1.c8.s4.39

The first digit of the multiplicand (5) is taken and multiplied with the digits of the multiplier. Thus 5× 2=10; 0 is written below 2, and 1 is to be carried over.¹ Then 5× 1=5; adding 1 (carried over), we get 6. 5 which is no longer required is rubbed out and 6 written in its place. Thus we have

v1.c8.s4.40

¹ For this purpose it was probably noted in a separate portion of the *pâṭî* by the beginner.

v1.c8.s4.41

The multiplier is then moved one place towards the left, and we have

v1.c8.s4.42

Now, 12 is multiplied by 3. The details are: 3× 2=6; this 6 added to the figure 6 below 2 gives 12. 6 is rubbed out and 2 substituted in its place. 1 is carried over. Then 3× 1=3; 3 plus 1 (carried over)=4. 3 is rubbed out and 4 substituted. After the multiplier 12 has been moved another place towards the left, the figures on the *pâṭî* stand thus:

v1.c8.s4.43

Then, 1× 2=2; 2+4=6; 4 is rubbed out and 6 substituted. 1× 1=1, which is placed to the left of 6.

v1.c8.s4.44

As the operation has ended, 12 is rubbed out and the *pāṭī* has

v1.c8.s4.45

Thus the numbers 12 and 135 have been *killed*¹ and a new number 1620 is born (*pratyutpanna*).²

v1.c8.s4.46

¹ This explains the use of the term *hanana* (killing) and its synonyms for multiplication.

v1.c8.s4.47

² Hence the product was termed *pratyutpanna*.

v1.c8.s4.48

The reader will note that the position of the multiplier and its motion serve two important purposes, *viz.*, (*i*) the last figure of the multiplier indicates the digit of the multiplicand by which multiplication is to be performed and, (*ii*) the product is to be added to the number standing underneath the digit of the multiplier multiplied.

v1.c8.s4.49

Sometimes the product of a digit of the multiplicand and the multiplier extends beyond the last place of the multiplier. In such cases, the last figure of the partial product is noted separately. The reader should note this fact in the case, 135 × 99, by performing the operation according to the above process.

v1.c8.s4.50

The beginner was liable to commit mistakes in such cases, (*i*) of not correctly taking into account the separately noted number, or (*ii*) of rubbing out the digit of the multiplicand beyond the last digit of the multiplier. For these reasons, this process was not in general use and the inverse process was preferred.

v1.c8.s4.51

*Inverse Method:* There appear to have been two varieties of the inverse method.

v1.c8.s4.52

(*a*) In the first the numbers are written thus: $12 / 135$ Multiplication begins with the last digit of the multiplicand. Thus 1 × 2 = 2; 1 is rubbed out and 2 substituted; then 1× 1=1, this is written to the left;¹ the multiplier 12 is moved to the next figure. The work on the *pâṭî* stands thus:

v1.c8.s4.53

¹ Or the alternative plan: 1× 1=1 and then 1× 2=2, thus giving 12 in the place of 1 in the multiplicand, etc.

v1.c8.s4.54

Then, 3× 2=6; 3 is rubbed out and 6 substituted; then 3× 1=3 and 3+2=5; 2 is rubbed out and 5 substituted in its place. The multiplier having been moved, the work on the *pâṭî* stands thus:

v1.c8.s4.55

Now, 5× 2=10; 5 is rubbed out and 0 substituted in its place; then 5× 1=5; 5+1=6; 6+6=12; 6 is rubbed out and 2 substituted, and 1 is carried over; then 1+5=6, 5 is rubbed out and 6 substituted in its place. The *pâṭî* has now,

v1.c8.s4.56

as the product (*pratyutpanna*). The figures to be carried over are noted down on a separate portion of the *pâṭî* and rubbed out after addition.

v1.c8.s4.57

(*b*) In the second the partial multiplications (*i.e.*, the multiplications by the digits of the multiplicand) are carried out in the direct manner. These partial multiplications, however, seem to have been carried out in the inverse way, this being the general fashion. The following example will illustrate the method of working:

v1.c8.s4.58

*Example.* Multiply 324 by 753

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The multiplier and the multiplicand are arranged thus:

v1.c8.s4.60

Multiplication begins with the last place of the multiplier. 3 × 7 gives 21; 1 is placed below the 7 of the multiplier and 2 to its left, thus:

v1.c8.s4.61

Then 3 × 5 gives 15; 5 is placed below the 5 of the multiplier and 1 carried to the left; the 1 obtained in the previous step is rubbed out and (1 + 1) = 2 is substituted, giving

v1.c8.s4.62

Then 3 × 3 gives 9; the 3 of the multiplicand is rubbed out and 9 substituted. The work on the *pāṭī* now stands thus:

v1.c8.s4.63

The multiplier is now moved one place to the right giving

v1.c8.s4.64

Then multiplying 7 by 2 we get 14. This 14 being set below the 7 gives

v1.c8.s4.65

Multiplying 5 by 2 and setting the result below it, we obtain

v1.c8.s4.66

Finally multiplying 3 by 2 and rubbing out 2, which is required no longer, and substituting 6 in its place, we get The multiplier is then moved one step further giving 753 / 240964 Multiplying by 4 the digits of the multiplier 753, and setting the results as before we obtain {lcl} (i) & {r} 753 / 243764 & multiplying 7 × 4 and setting the result; / (ii) & {r} 753 / 243964 & multiplying 5 × 4 and setting the result; / (iii) & {r} 753 / 243972 & multiplying 3 × 4 and setting the result.

v1.c8.s4.67

It may be again remarked that the position and motion of the multiplier play a very important part in the above process. The digits of the multiplier are also successively rubbed out in order to avoid confusion, thus 7 is rubbed out at stage *(i)*, 5 at *(ii)* and 3 at *(iii)*.

v1.c8.s4.68

The following variation of the above process is also found:¹

v1.c8.s4.69

¹ *Līlāvatīyudāharaṇa* by Kṛpārâma Daivajña, Asiatic Society of Bengal, Calcutta, Ms. No. III. F. 110. A.

v1.c8.s4.70

"Multiplicand 135, multiplier 12; the multiplier placed at the last place of the multiplicand gives $12 / 135$ According to the rule 'the numerals progress to the left' the last figure of the multiplicand (the figure 1) is multiplied by 12. Then after moving (12) we get $12 / 1235$

v1.c8.s4.71

Again, the figure 3 next to the last of the multiplicand being multiplied by the multiplier 12 gives $12 / 1265 / 3$

v1.c8.s4.72

Then after moving (12) we get $12 / 1265 / 3$ Again, multiplying the first figure 5 of the multiplicand with the multiplier 12, we get $12 / 1260 / 36$ Then rubbing out the multiplier, the numbers $1260 / 36$ being added according to places give 1620."

v1.c8.s4.73

**Transmission to the West.** The *kapâṭa-sandhi* method of multiplication was transmitted to the Arabs who learnt the decimal arithmetic from the Hindus. It occurs in the works of Al-Khowârîzmî (825), Al-Nasavî¹ (*c.* 1025) Al-Haṣṣâr² (*c.* 1175), Al-Kalasâdî³ (*c.* 1475) and many others. The following illustration is taken from the work of Al-Nasavî who calls this method *al-amal al-hindî* and *târîk al-hindî* ("the method of the Hindus"):

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¹ F. Woepcke, I (6), p. 407.

v1.c8.s4.75

² H. Suter, *Bibl. Math.*, II (3), p. 16.

v1.c8.s4.76

³ *Ibid*, p. 17.

v1.c8.s4.77

*Example.* To multiply 324 × 753 $43 / /309 / /2/977 / 21/5//9/62 & ∴ Product = 243972. / /3/24 / 753 / 753 / 753$

v1.c8.s4.78

In the above the arrangement of the multiplicand and multiplier is just the same as in the Hindu method. The multiplier is moved in the same way. As the work is performed on paper, the figures are crossed out instead of being rubbed out.

v1.c8.s4.79

It may be mentioned that in Europe, the method is found reproduced in the work of Maximus Planudes.

v1.c8.s4.80

**Gelosia Method.** The method known as the 'gelosia',¹ has been described in the *Gaṇita-mañjarî* (16th century) as the *kapâṭa-sandhi* method. It appears also in Gaṇeśa's commentary on the *Lîlâvatî*. As the description of the *kapâṭa-sandhi* given by the older mathematicians is incomplete and sketchy, it is difficult to say whether Gaṇeśa is right in identifying the gelosia method with the *kapâṭa-sandhi* of older writers. In our opinion Gaṇeśa's identification is incorrect.²

v1.c8.s4.81

¹ We shall designate it as *kapâṭa-sandhi* (*b*) method.

v1.c8.s4.82

² *Cf.* the quotation from Śrîpati given before, p. 137.

v1.c8.s4.83

We are at present unable to say definitely whether this method is a Hindu invention or was borrowed from the Arabs who are said to have used it in the 13th century.³ It occurs in some Arab works of the 14th century, and also in Europe about the same time. Gaṇeśa was undoubtedly one of the best mathematicians of his time and the fact that he identified this method with the *kapâṭa-sandhi* which is the oldest known method shows that the gelosia method must have been in use in India from a long time before him.

v1.c8.s4.84

³ Smith, *History*, II, p. 115.

v1.c8.s4.85

The only available description of the method runs as follows:

v1.c8.s4.86

"(Construct) as many compartments as there are places in the multiplicand and below these as many

v1.c8.s4.87

as there are places in the multiplier; the oblique lines in the first, in the one below, and in the other (compartments) are produced. Multiply each place of the multiplicand, by the places of the multiplier (which are) one below the other and set the results in the compartments. The sum taken obliquely on both sides of the oblique lines in the compartments gives the product. This is the *kapâṭa-sandhi*."¹

v1.c8.s4.88

¹ Translated from the *Gaṇita-mañjarî* of Gaṇeśa, son of Dhuṇḍhirâja.

v1.c8.s4.89

The following illustration is taken from Gaṇeśa's commentary on the *Lîlâvatî*:

v1.c8.s4.90

To Multiply 135 by 12

v1.c8.s4.92

**Cross Multiplication Method.** This method has been mentioned by Śrîdhara, Mahâvîra, Śrîpati and some later writers as the *tastha* method. These writers, however, do not explain the method. Śrîdhara simply states: "The next (method) in which (the multiplier) is stationary is the *tastha*."² The method is algebraic and has been compared to *tiryak-guṇana* or *vajrâbhyâsa* (cross multiplication) used in algebra.³ It has been explained by Gaṇeśa (*c.* 1545) thus:

v1.c8.s4.93

² *Triś*, p. 3.

v1.c8.s4.94

³ Colebrooke, *l.c.*, p. 171, fn. 5.

v1.c8.s4.95

“That method of multiplication in which the numbers stand in the same place,¹ is called *tastha-guṇana*. It is as follows: after setting the multiplier under the multiplicand multiply unit by unit and note the result underneath. Then as in *vajrâbhyâsa* multiply unit by ten and ten by unit, add together and set down the result in the line. Next multiply unit by hundred, hundred by unit and ten by ten, add together and set down the result as before; and so on with the rest of the digits. This being done, the line of results is the product.”²

v1.c8.s4.96

¹ In contra-distinction to the method in which the multiplier moves from one place to another.

v1.c8.s4.97

² Gaṇeśa’s commentary on the *Lîlâvatî*, i, 4-6.

v1.c8.s4.98

This method was known to the Hindu scholars of the 8th century, or earlier. The method seems to have travelled to Arabia and thence was transmitted to Europe, where it occurs in Pacioli’s *Suma*³ and is stated to be “more fantastic and ingenious than the others.” Gaṇeśa has also remarked that “this (method) is very fantastic and cannot be learnt by the dull without the traditional oral instructions.”

v1.c8.s4.99

³ Smith (*l.c.*, II, p. 112) quotes from this work.

v1.c8.s4.100

**Multiplication by Separation of Places.** This method of multiplication known as *sthâna-khaṇḍa*, is based on the separation of the digits of the multiplicand or of the multiplier. It has been mentioned in all the works from 628 A.D. onwards. Bhâskara II describes the method as follows:

v1.c8.s4.101

“Multiply separately by the places of figures and add together.”⁴

v1.c8.s4.102

⁴ *L*, p. 3.

v1.c8.s4.103

With reference to the example 135 × 12, Bhâskara II explains the method thus:

v1.c8.s4.104

"Taking the digits separately, *viz.*, 1 and 2, the multiplicand being multiplied by them severally, and the products added together according to places, the result is 1620."

v1.c8.s4.105

Various arrangements appear to have been employed for writing down the working. Some of these are given below:

v1.c8.s4.106

(i)¹ {r} 135 / 12 / \hline 12\phantom / 36\phantom / 60 / \hline 1620

v1.c8.s4.107

¹ In a manuscript used by Taylor, see his *Lîlâwatî*, pp. 8-9.

v1.c8.s4.108

(ii)² {ccc} 12 & 12 & 12 / 1 & 3 & 5 / \hline \multicolumn{c} / \multicolumn{c} / \hline \multicolumn{c}

v1.c8.s4.109

² This arrangement is found in the commentary of Gaṅgâdhara on the *Lîlâvatî*, in the library of the Asiatic Society of Bengal, Calcutta.

v1.c8.s4.110

(iii)³ {cc} 135 & 135 / 1 & 2 / \hline \multicolumn{c} / \multicolumn{c}{135\phantom} / \hline \multicolumn{c}

v1.c8.s4.111

³ Found in Gaṅgâdhara, *l.c.*

v1.c8.s4.112

**Zigzag Method.** The method is called *gomûtrikâ*.⁴ It has been described by Brahmagupta. It is in all essentials the same as the *sthāna-khaṇḍa* method. The following illustration is based on the commentary of Pṛthudakasvâmî.

v1.c8.s4.113

⁴ The word *gomûtrikâ*, means "similar to the course of cow's urine," hence "zigzag." Colebrooke's reading *gosûtrikâ* is incorrect. The method of multiplication of astronomical quantities is called *gomûtrikâ* even upto the present day by the paṇḍits.

v1.c8.s4.114

*Example.* To multiply 1223 by 235

v1.c8.s4.115

The numbers are written thus :

v1.c8.s4.116

{cc} 2 & 1223 / 3 & 1223 / 5 & 1223

v1.c8.s4.117

The first line of figures is then multiplied by 2, the process beginning at the units place, thus: 2 × 3 = 6; 3 is rubbed out and 6 substituted in its place, and so on. After all the horizontal lines have been multiplied by the corresponding numbers on the left in the vertical line, the numbers on the *pâṭî* stand thus:

v1.c8.s4.118

{cccccc} 2 & 4 & 4 & 6 & & / & 3 & 6 & 6 & 9 & / & & 6 & 1 & 1 & 5 / \hline 2 & 8 & 7 & 4 & 0 & 5

v1.c8.s4.119

after being added together as in the present method.

v1.c8.s4.120

The *sthāna-khaṇḍa* and the *gomūtrikā* methods resemble the modern plan of multiplication most closely. The *sthāna-khaṇḍa* method was employed when working on paper.

v1.c8.s4.121

**Parts Multiplication Method.** This method is mentioned in all the Hindu works from 628 A.D. onwards. Two methods come under this head:

v1.c8.s4.122

(*i*) The multiplier is broken up into two or more parts whose sum is equal to it. The multiplicand is then multiplied severally by these and the results added.¹

v1.c8.s4.123

¹ Thus 12 × 135 = (4 + 8) × 135 = (4 × 135) + (8 × 135).

v1.c8.s4.124

(*ii*) The multiplier is broken up into two or more aliquot parts. The multiplicand is then multiplied by one of these, the resulting product by the second and so on till all the parts are exhausted. The ultimate product is the result.¹

v1.c8.s4.125

¹ Thus 12× 135 = 3× 135× 4.

v1.c8.s4.126

These methods are found among the Arabs and the Italians, having been obtained from the Hindus. They were known as the "Scapezzo" and "Repiego" methods respectively among the Italians.²

v1.c8.s4.127

² Smith, *History*, II, p. 117.

v1.c8.s4.128

**Algebraic Method.** This method was known as *iṣṭa-guṇana*. Brahmagupta's description of the method has been already quoted. Bhâskara II explains it thus:

v1.c8.s4.129

"Multiply by the multiplicator diminished or increased by an assumed number, adding or subtracting (respectively) the product of the multiplicand and the assumed number."³

v1.c8.s4.130

³ *L*, p. 3.

v1.c8.s4.131

This is of two kinds according as we (*i*) add or (*ii*) subtract an assumed number. The assumed number is so chosen as to give two numbers with which multiplication will be easier than with the original multiplier. The two ways are illustrated below:

v1.c8.s4.132

(i) 135× 12&=135×(12+8)-135× 8 / &=2700-1080=1620

v1.c8.s4.133

(ii) 135× 12&=135×(12-2)+135× 2 / &=1350+270=1620

v1.c8.s4.134

This method was in use among the Arabs⁴ and in Europe⁵, obviously under Hindu influence.

v1.c8.s4.135

⁴ *E.g.*, Behâ Eddin (*c.* 1600). See G. Eneström, *Bibl. Math.*, VII (3), p. 95.

v1.c8.s4.136

⁵ *E.g.*, Widman (1489), Riese (1522), etc. See Smith, *l.c.*, p. 120.

5. Division

v1.c8.s5.1

**Terminology.** Division seems to have been regarded as the inverse of multiplication. The common Hindu names for the operation are *bhâgahâra*, *bhâjana*, *haraṇa*, *chedana*, etc. All these terms literally mean "to break into parts," *i.e.*, "to divide," excepting *haraṇa* which denotes "to take away." This term shows the relation of division to subtraction. The dividend is termed *bhâjya*, *hârya*, etc., the divisor *bhâjaka*, *bhâgahara* or simply *hara*, and the quotient *labdhi* "what is obtained" or *labdha*.

v1.c8.s5.2

**The Operation.** Division was considered to be a difficult and tedious operation by European scholars even as late as the 15th and 16th centuries;¹ but in India the operation was not considered to be difficult, as the most satisfactory method of performing it had been evolved at a very early period. In fact, no Hindu mathematician seems to have attached any great importance to this operation. Āryabhaṭa I does not mention the method of division in his work. But as he has given the modern methods for extracting square- and cube-roots, which depend on division,² we conclude that the method of division was well-known in his time and was not described in the *Āryabhaṭīya* as it was considered to be too elementary. Most Siddhânta writers have followed Āryabhaṭa in excluding the process of division from their works, *e.g.*, Brahmagupta (628), Śrîpati (1039), and some others.

v1.c8.s5.3

¹ Smith, *l.c.*, p. 132.

v1.c8.s5.4

² He has used the technical term *labdha* for the quotient.

v1.c8.s5.5

A method of division by removing common factors seems to have been employed in India before the invention of the modern plan. This removal of common factors is mentioned in early Jaina works.¹ It has been mentioned by Mahâvîra who knew the modern method, probably because it was considered to be suitable in certain particular cases:

v1.c8.s5.6

¹ *Tattvârthâdhigama-sûtra*, *Bhâṣya* of Umâsvâti (*c.* 160, ed. by H. R. Kapadia, Bombay, 1926, Part I, ii. 52, p. 225.

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