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History of Hindu Mathematics: A Source Book v1.c7.s3: 3. Scope and Development of Hindu Mathematics

History of Hindu Mathematics: A Source Book v1.c7.s3

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3. Scope and Development of Hindu Mathematics

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*Gaṇita* literally means "the science of calculation" and is the Hindu name for mathematics. The term is a very ancient one and occurs copiously in Vedic literature. The *Vedâṅga Jyotiṣa* (*c.* 1,200 B.C.) gives it the highest place of honour among the sciences which form the *Vedâṅga*: "As the crests on the heads of peacocks, as the gems on the hoods of snakes, so is *gaṇita* at the top of the sciences known as the *Vedâṅga*."¹ In ancient Buddhist literature we find mention of three classes of *gaṇita*: (1) *mudrâ* ("finger arithmetic"), (2) *gaṇanâ* ("mental arithmetic") and (3) *saṁkhyâna* ("higher arithmetic in general"). One of the earliest enumerations of these three classes occurs in the *Dîgha Nikâya*,² and it is also found in the *Vinaya Piṭaka*,³ *Divyâvadâna*⁴ and *Milindapañho*.⁵ The word *saṁkhyâna* has been used for *gaṇita* in several old works.⁶ At this remote period *gaṇita* included astronomy, but geometry (*kṣetra-gaṇita*) belonged to a different group of sciences known as *Kalpasûtra*.

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¹ "Yathâ śikhâ mayurâṇâṁ nâgânâṁ maṇayo yathâ Tadvadvedâṅgaśâstrâṇâṁ gaṇitaṁ mûrdhani sthitam." *Vedâṅga Jyotiṣa*, 4.

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² I, p. 51.

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³ IV, p. 7.

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⁴ *Divyâvadâna*, ed. by E. B. Cowell and R. A. Neil, Cambridge, 1886, pp. 3, 26 and 88.

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⁵ *Milindapañho*, Eng. trans. by Rhys Davids, Oxford, 1890, p. 91.

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⁶ E.g., *Kalpasûtra* of Bhadrabâhu, ed. by H. Jacobi, Leipzig, 1897; *Bhagavatî-sûtra*, Bombay, 1918, p. 112; *Arthaśâstra*, i. 5. 2.

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It is believed that some time before the beginning of the Christian era, there was a renaissance of Hindu *Gaṇita*.⁷ The effect of this revival on the scope of *gaṇita* was great. Astronomy (*jyotiṣa*) became a separate subject and geometry (*kṣetra-gaṇita*) came to be included within its scope. The subjects treated in the Hindu *Gaṇita* of the early renaissance period consisted of the following:¹ *Parikarma* ("fundamental operations"), *Vyavahâra* ("determinations"), *Rajju* ("rope," meaning geometry), *Râśi* ("rule of three"), *Kalâsavarṇa* ("operations with fractions"), *Yâvat tâvat* ("as many as," meaning simple equations), *Varga* ("Square," meaning quadratic equations), *Ghana* ("Cube", meaning cubic equations), *Varga-varga* (biquadratic equations) and *Vikalpa* ("permutations and combinations").

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⁷ Bibhutibhusan Datta, "The scope and development of Hindu Gaṇita," *Indian Historical Quarterly*, V, 1929, pp. 479-512.

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¹ "Parikammaṁ vavahâro rajju râsî kalâsavanne ya । Jâvantâvati vaggo ghano tataha vaggavaggo vikappo ta ॥" *Sthânâṅgasûtra*, Sûtra 747.

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Thus *gaṇita* came to mean mathematics in general, while 'finger arithmetic' as well as 'mental arithmetic' were excluded from the scope of its meaning For the calculations involved in *gaṇita*, the use of some writing material was essential. The calculations were performed on a board (*pâṭî*) with a piece of chalk or on sand (*dhûlî*) spread on the ground or on the *pâṭî*. Thus the terms *pâṭî-gaṇita* ("science of calculation on the board") or *dhûlî-karma* ("dust-work"), came to be used for higher mathematics. Later on the section of *gaṇita* dealing with algebra was given the name *Bîja-gaṇita*. The first to effect this separation was Brahmagupta (628), but he did not use the term *Bîja-gaṇita*. The chapter dealing with algebra in his *Brâhma-sphuṭa-siddhânta* is called *Kuṭṭaka*. Śrîdharâcârya (750) regarded *Pâṭî-gaṇita* and *Bîja-gaṇita* as separate and wrote separate treatises on each. This distinction between *Pâṭîgaṇita* and *Bîjagaṇita* has been preserved by later writers.

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Having given a brief survey of the position and scope of mathematics in Ancient India, we turn to the purpose in hand—that of giving a connected account of the development and growth of the different branches of mathematics. The numeration system of the Hindus will engage our attention first.

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