History of Hindu Mathematics: A Source Book v1.c8.s4: 4. Multiplication

History of Hindu Mathematics: A Source Book v1.c8.s4

Language: English

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.

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*Example*. To multiply 135 by 12.

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

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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

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

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

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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:

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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.

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

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

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

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² Hence the product was termed *pratyutpanna*.

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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.

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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.

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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.

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

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(*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:

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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.

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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:

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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,

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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.

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(*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:

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*Example.* Multiply 324 by 753

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The multiplier and the multiplicand are arranged thus:

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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:

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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

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

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The multiplier is now moved one place to the right giving

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

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

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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.

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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)*.

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

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

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"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 $12 / 1235$

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

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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."

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**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.

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

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³ *Ibid*, p. 17.

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

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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.

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

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**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

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

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

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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.

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³ Smith, *History*, II, p. 115.

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The only available description of the method runs as follows:

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

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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

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

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

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To Multiply 135 by 12

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**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:

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² *Triś*, p. 3.

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³ Colebrooke, *l.c.*, p. 171, fn. 5.

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“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

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

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

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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.”

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

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**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:

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

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⁴ *L*, p. 3.

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

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"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."

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

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

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

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

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

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

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³ Found in Gaṅgâdhara, *l.c.*

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**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.

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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.

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*Example.* To multiply 1223 by 235

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The numbers are written thus :

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

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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:

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

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after being added together as in the present method.

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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.

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

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(*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

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

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(*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

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¹ Thus 12× 135 = 3× 135× 4.

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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

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² Smith, *History*, II, p. 117.

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

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"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

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³ *L*, p. 3.

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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:

Linked texts (9)

Āryabhaṭīya — Āryabhaṭa, Sanskrit
Gaṇitasārasaṅgraha — Mahāvīra (mathematician), Sanskrit
Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Siddhāntaśekhara — Śrīpati, Sanskrit
Bakhshālī Manuscript — Sanskrit
Mahāsiddhānta (Āryasiddhānta) — Āryabhaṭa II, Sanskrit
Gaṇitatilaka — Śrīpati, Sanskrit
Triśatikā (Pāṭīgaṇitasāra) — Śrīdhara (mathematician), Sanskrit