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A Concise History of Science in India c7.s2.t2: Arithmetic

A Concise History of Science in India c7.s2.t2

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English

Arithmetic

c7.s2.t2.1

*Number*

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Like the Egyptians, the Vedic Hindus adopted 10 as the basis of numeration and developed a great interest in thinking out and naming large numbers. The Greeks, on the other hand, fought shy of large numbers and their terminology hardly took them beyond the ‘myriad’ (10⁴). The Egyptians freely dealt with large numbers on the decimal scale, such as hundred, thousand, million, tens of million and so on and had meaningful Hieroglyphic symbols to express them. The same is true of the Vedic Hindus. The various recensions of the *Yajurveda Saṃhitā* give names to numbers as large as 10¹². The *Taittirīya Saṃhitā*a gives the following denominations: *eka* (1), *daśa* (10), *śata* (10²), *sahasra* (10³), *ayuta* (10⁴), *niyuta* (10⁵), *prayuta* (10⁶), *arbuda* (10⁷), *nyarbuda* (10⁸), *samudra* (10⁹), *madhya* (10¹⁰), *anta* (10¹¹) and *parārdha* (10¹²). The same is repeated in the *Pañcaviṃśa Brāhmaṇa* with further extensions. The following passage from the *Pañcaviṃśa Brāhmaṇa*b will give an idea of the context and the manner in which large numbers were introduced:

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‘By sacrificing with the *agnihotra*, he reaches in one day ten “House-lords” (i.e. he becomes equal to, gains the merits of ten H.); by sacrificing with the *agnihotra* during ten years, he becomes equal to one who (regularly) performs the sacrifices of full and new moon, he becomes equal to one who performs the sacrifices of *soma*. By offering the *agniṣṭoma* sacrifices, he becomes equal to one who performs a sacrifice of a thousand cows as sacrificial fee. By offering ten of these, he becomes equal to one who performs a sacrifice with ten thousand *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of a hundred thousand *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of a million *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of 10 million *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of 100 million *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of 1,000 million *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of 10,000 million *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of 100,000 million *dakṣiṇās*. By offering ten of these he becomes the cow.’

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The method of obtaining higher and higher numbers in multiples of 10, described later on as *daśaguṇottara*, is clearly indicated. The number names in this decimal scale in ascending order are: *eka*, *daśa*, *śata*, *sahasra*, etc., the same as given in the *Taittirīya Saṃhitā*. It appears that the thinking out and the naming of such large numbers were a favourite pastime of the ancient Indian mathematicians. The same tendency is noticed in the Buddhist literature where we find a centesimal scale (*śatottara gaṇanā*) and the name *tallakṣaṇa* for the number 10⁵⁸. The Jaina work, the *Anuyo-gadvāra-sūtra* (c. 100 B.C.), called the places in powers of 10 as *gaṇanā-sthāna* and mentioned large numbers up to 29 places and beyond. The Jainas are further credited with the conception of a time-scale called *śīrṣa-prahelikā* (8,400,000) and the suggestion of building up fantastically large numbers in ascending powers of this figure.

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a *Taitt. S.*, 4.4.11.4.; 7.2.20.1.

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b *Pañc. Br.*, xvii. 14.1.2; Eng. trans. by Caland.

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The Vedic Hindus showed the same proficiency in developing a scientific vocabulary of number names, in which the principles of addition, subtraction and multiplication were conveniently used. The system required the naming of (1) the first nine digits, e.g. *eka*, *dvi*, *tri*, *catur*, *pañca*, *ṣaṭ*, *sapta*, *aṣṭa* and *nava*; (2) the second group of nine numbers obtained by multiplying each of the above digits by 10, e.g. *daśa*, *viṃśati*, *triṃśat*, *catvāriṃśat*, *pañcāśat*, *ṣaṣṭi*, *saptati*, *aśīti* and *navati*; and (3) the third group of 11 numbers beginning with 100 and followed by its multiples of 10 and its higher powers, e.g. *śata*, *sahasra*, *ayuta*, . . . *parārdha* (see above). The multiplicative principle is already indicated in forming the numbers of the second and third group. Other examples are *ṣaṣṭiṃ sahasrāṇi*, *pañcāśat sahasram*, *dvā-saptatiḥ sahasrāṇi*. The additive principle is generally used in naming a number in which the numbers of the first and second group participate, e.g. *eka-daśa* (11), *sapta-viṃśati* (27), *aṣṭa-triṃśat* (38). Additive and multiplicative principles are simultaneously used when, in the number concerned, the members of the third group participate along with those of the second or the first, e.g. *sapta śatāni viṃśatiḥ* (720), *ṣaṣṭiṃ-sahasra navatim nava* (60,099).a These principles were widely used in constructing number names in other culture areas, as witness such terms as *pente-deka* (50), *pente-hekaton* (500) among the Greeks.

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Of greater importance, however, is the infrequent application of the subtractive principle for which the Etruscans are generally given the credit.b Proceeding from right to left, they wrote 27 as XIIIXX [20+(10−3)]; 38 as XIIXXX [30+(10−2)] and so on. The same subtractive principle, in the number nomenclature, was applied by the Vedic Hindus in the coinage of such terms as *ekānna-śata* (100−1), *ekānna-viṃśati* (20−1), *ekānna-catvāriṃśat*,c etc. The word *ekānna*, meaning ‘one less’, was later on shortened into *ekona* and *ūna*. In a limited way, the Babylonians also used this principle in forming their inauspicious number 19 as <<ᵞ< (20−1), where ᵞ stands for −1.

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Fundamental Operations

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The four fundamental arithmetical operations, e.g. addition, subtraction, multiplication and division, are nowhere spelled out in the Vedic literature, as these were obviously taken for granted being commonplace operations. One, however, meets with the interesting case of dividing one thousand into three equal parts in the *Ṛgveda* and the *Brāhmaṇas*. In the *Ṛgveda*, the division of the thousand is not explicitly mentioned, but it is done so in the *Tattirīya Saṃhitā*d which, while incorporating the *ṛc*, modifies the last line as ‘Ye did divide the thousand into three’. The full explanation of the arithmetical feat of Indra and Viṣṇu in the proposed division is given in the *Śatapatha Brāhmaṇa* as follows: 'For when Indra and Viṣṇu divided a thousand (cows) into three parts, there was one left, and here they caused to propagate herself in three kinds; and hence, even now, if anyone were to divide a thousand by three, one would remain over.'a

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a *RV.* I. 164.11; I. 53.9.

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b Smith (D. E.), II, p. 58.

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c *Taitt. S.*, vii. 2.11.

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d *Taitt. S.*, vi. 1.6.

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*Fractions*

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Acquaintance with the fundamental arithmetical operations with elementary fractions is clearly indicated in the Vedic texts and their appendages. The *Ṛgveda* gives names of a number of simple fractions such as *ardha* (½), *tripāda* (¾) and the *Maitrāyaṇī Saṃhitā* mentions *pāda* (¼), *śapha* (⅛), *kuṣṭha* (1/12), *kalā* (1/16). From the *Śulba-sūtras* onwards we meet with terms such as *aṃśa*, *bhāga* to denote fractions in general. These terms are used in combination with cardinal or ordinal number names; in the case of the latter the word *aṃśa* or *bhāga* is often omitted. A few examples follow:

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{ll} tribhāga, tryaṃśa & = 1/3 / pañcama-bhāga, pañcama & = 1/5 / dvādaśa-bhāga or dvādaśa & = 1/12 / pañcadaśa-bhāga & = 1/15 / tri-aṣṭama, tryaṣṭa & = 3/8 / dvi-saptama & = 2/7 / trayastryaṣṭa & = 3 3/8 / pañcamasya caturviṃśa & = 1/24 of 1/5

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There are a number of other peculiar modes of expressing fractions, e.g. *ardhāṣṭama* (7½), *ardhanavama* (8½), *ardhadaśama* (9½)b or *dvi-guṇa* (½), *tri-guṇa* (⅓), *caturguṇa* (¼).c

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The *Śulba-sūtras* contain several instances of addition, subtraction, multiplication, division and the squaring of fractions. Consider the following example from Baudhāyana:d

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'Then he measures the area of this square-shaped *citi*, whose side is three *puruṣas* less one-third. At the western side of this square is the handle whose east-west length is half a *puruṣa* plus ten *aṅgulas* (one-twelfth *puruṣa*) and north-south breadth one *puruṣa* less one-third. This makes the (area of the) *agni* sevenfold plus two *aratnis* and the *prādeśa*.'

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c *Mśl.*, v. 5.

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Two *aratnis* and the *prādeśa* equal 60 *aṅgulas* or ½ *puruṣa*. Expressed in figures, the above means:

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b *Bśl.*, ii. 1–3.

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d *Bśl.*, iii. 219–24.

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The area of the *agni* =(3-⅓)²+(½+1/12)(1-⅓)=7½.

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a *Śat. Br.*, iii. 3.1.13.

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The results of a few more statements of this nature are given below:

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(1) Number of bricks each 1/25 sq. *puruṣa* to cover 7½ sq. *puruṣa*
= 7½ ÷ 1/25 = 187½
(2) Number of bricks each 1/15 of ½ sq. *puruṣa* to cover 7½ sq. *puruṣa*
= 7½ ÷ 1/15 of ½ = 15/2 ÷ 1/30 = 15/2 × 30 = 225.

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*Progressive Series*

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Indian interest in progressive series has been traced to the *Saṃhitās*. The *Taittirīya Saṃhitā* indicates the following arithmetical series in which the odd (*ayugma*) and the even (*yugma*) ones are noticeable:

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1, 3, 5, ... 19, 29, 39, ... 99
2, 4, 6, 20
4, 8, 12, 20
5, 10, 15, 100
10, 20, 30, 100

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The *Pañcaviṃśa Brāhmaṇa*a prescribes *dakṣiṇās* in gold measures (*mānas*) to be given in geometrical progressions, depending on the nature and the time of offerings, as follows:

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12 (gold *mānas*), 24, 48, 96, 192 ... 3072, 6144 ... 49152, 98304, 196608, 393216.

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In the above examples, the summations of such series are not given. This is correctly done in the *Śatapatha Brāhmaṇa*b as follows: ‘... there are seven of these meters, increasing by four syllables; and the triplets of these make seven hundred and twenty syllables and thirty-six in addition thereto.’

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In other words, it is proposed to compute

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3(24+28+32+ 7 terms) = 3 × 7/22× 24+(7-1)× 4 = 756.

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It is true that no general rule for finding the summation as indicated above is given in the *Śatapatha Brāhmaṇa*. But the recurrence of both arithmetical and geometrical series, with the correct statement of the results of their summation, strongly suggests that the Vedic Hindus probably possessed some method of finding the summation of such series. Thus the *Bṛhaddevatā* (400 B.C.) gives the summation

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2+3+4+ +1000 = 500499

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and Bhadrabāhu’s *Kalpasūtra* (300 B.C.), a Jaina canonical work, the summation of the geometrical series

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b *Śat. Br.*, x. 5.4.7

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1+2+4+ +8192 = 16383.

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The series played an important part in prosody. Piṅgala, in his *Chandaḥsūtra* (200 B.C.), used a geometrical series 2, 2², 2³

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a *Pañc. Br.*, xviii. 3.

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It is noteworthy that the summation of progressive series has ever since continued to occupy the attention of subsequent Indian mathematicians such as Mahāvīra, Bhāskara II, Nārāyaṇa and others who have all given general formulas for obtaining the sum. In the *Sthānāṅgasūtra*, progressive series is included in the mathematical topic *Vyavahāra*, as explained by the commentator Abhayadeva Sūrī (A.D. 1050). Later on, it became the subject of special treatises, *e.g.* *Bṛhaddhāra Parikarma*, referred to by Nemicandra.

Cites (6)