History of Hindu Mathematics: A Source Book v1.c8.s5: 5. Division
History of Hindu Mathematics: A Source Book v1.c8.s5
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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.
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¹ Smith, *l.c.*, p. 132.
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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:
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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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"Putting down the dividend and below it the divisor, and then, having performed division by the method of removing common factors, give out the resulting (quotient)."²
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² *GSi*, p. 11. The method would not give the quotient unless the dividend be completely divisible by the divisor.
v1.c8.s5.9
The modern method of division is not found in the Bakhshâlî Manuscript, although the name of the operation is found at several places. The absence of the method may be due to the mutilated form of the text, although it is quite possible that the method was not known at that early period (200 A.D.).
v1.c8.s5.10
**The Method of Long Division.** The modern method of division is explained in the works on *pâṭîgaṇita*, the earliest of which, Śrîdhara's *Triśatikâ*, gives the method as follows:³
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³ *Triś*, p. 4.
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"Having removed the common factor, if any, from the divisor and the dividend, divide by the divisor (the digits of the dividend) one after another in the inverse⁴ order."
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⁴ *Pratiloma*.
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Mahâvîra says:⁵
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⁵ *GSS*, p. 11; *cf.* Rangacarya's translation.
v1.c8.s5.16
"The dividend should be divided by the divisor (which is) placed below it, in the inverse order, after having performed on them the operation of removing common factors."
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Âryabhaṭa II gives more details of the process:¹
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¹ *MSi*, p. 144.
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“Perform division having placed the divisor below the dividend; subtract from (the last digits of the dividend) the proper multiple of the divisor; this (the multiple) is the partial quotient, then moving the divisor divide what remains, and so on.”
v1.c8.s5.20
Bhâskara II,² Nârâyaṇa³ and others give the same method.
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² Bhâskara gives the process briefly as follows: “That number, by which the divisor being multiplied, balances the last digit of the dividend gives the (partial) quotient, and so on.” (*L*, p. 3)
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³ *GK*, i. 16.
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The following example will serve to illustrate the Hindu method of performing the operation on a *pâṭî*:
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*Example.* Divide 1620 by 12.
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The divisor 12 is placed below the dividend thus : $1620 / 12$
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The process begins from the extreme left of the dividend, in this case the figure 16. This 16 is divided by 12. The quotient 1 is placed in a separate line, and 16 is rubbed out and the remainder 4 is substituted in its place. The subtraction is made by rubbing out figures successively as each figure of the product to be subtracted is obtained. Thus, the partial quotient 1, being written, the procedure is 1620 / 12 I/(line of quotients) 1 × 1 = 1, so 1 of the dividend is rubbed out (as 1 - 1 = 0); then 1 × 2 = 2, so 4 is substituted in the place of 6 (as 6 - 2 = 4). The figures on the *pâṭî* are: 420 / 12 I/(line of quotients)
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The divisor 12 is now moved one place to the right giving
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$420 & 1/(line of quotients) / 12 &$
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42 is then divided by 12. The resulting quotient 3 is set in the "line of quotients," 42 is rubbed out and the remainder 6 substituted in its place. The figures now stand thus:
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$60 & 13/(line of quotients) / 12 &$
v1.c8.s5.31
Moving the divisor one place to the right, we have
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On division being performed, as before the resulting quotient 5 is set in the "line of quotients" and 60 is rubbed out leaving no remainder. The line of quotients¹ has
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¹ The "line of quotients" was usually written above the dividend.
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which is the required result.
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The above process, when the figures are not obliterated and the successive steps are written down one below the other, becomes the modern method of long division.
v1.c8.s5.36
The method seems to have been invented in India about the 4th century A.D., if not earlier. It was transmitted to the Arabs, where it occurs in Arabic works from the 9th century onwards.² From Arabia the method travelled to Europe where it came to be known as the galley (*galea*, *batello*) method.³ In this variation of the method, the figures obtained at successive stages are written and crossed out, for the work is carried out on paper (where the figures cannot be rubbed out). The method was very popular in Europe from the 15th to the 18th century.¹ The above example worked on the galley plan would be represented thus:
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² Al-Khowârîzmî (*c.* 825), Al-Nasavî (*c.* 1025); cf. Smith, *l.c.*, pp. 138-139.
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³ Also called the 'scratch method'.
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¹ For details see Smith, *l.c.*, pp. 136-139.
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| | 4 | |
| I | 1\cancel20 | 1 |
| | \cancel\cancel2 | |
| | 1 | |
| | | |
| II | \cancel | |
| | 4\cancel | |
| | 1\cancel\cancel0 | 13 |
| | \cancel\cancel\cancel2 | |
| | \cancel1 | |
| | | |
| III | \cancel\cancel | |
| | 4\cancel | |
| | 1\cancel\cancel\cancel | 135 |
| | \cancel\cancel\cancel\cancel | |
| | \cancel | |
v1.c8.s5.41
Comparing the successive crossing out of the figures in I, II and III, with the rubbing out of figures in the corresponding steps according to the Hindu plan, it becomes quite clear that the galley method is exactly the same as the Hindu method. The crossing out of figures appears to be more cumbrous than the elegant Hindu plan of rubbing out.
v1.c8.s5.42
The Hindu plan of moving the divisor as the digits of the quotient were evolved, although not essential, was also copied and occurs in the works of such well-known Arab writers as Al-Khowârîzmî (825), Al-Nasavî (*c.* 1025) and others. The mediæval Latin writers called this feature the *antirioratio*.
Linked texts (6)
- Gaṇitasārasaṅgraha — Mahāvīra (mathematician), Sanskrit
- Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
- Mahāsiddhānta (Āryasiddhānta) — Āryabhaṭa II, Sanskrit
- Tattvārthasūtra — Umāsvāti, Sanskrit
- Triśatikā (Pāṭīgaṇitasāra) — Śrīdhara (mathematician), Sanskrit
- Gaṇitakaumudī — Nārāyaṇa Paṇḍita (mathematician), Sanskrit