Mathematics in Ancient and Medieval India c7.s4: Pascal Triangle and the Binomial Theorem

Mathematics in Ancient and Medieval India c7.s4

Language: English

Pascal Triangle and the Binomial Theorem

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The credit of expansion of Binomial theorem :

(a+b)n=an+nc_₁ an-1b+nc_₂ an-2 b²+..........+bn in its simplest form i.e., when n is a positive integer, goes to India and it is the Indian scholar Halāyudha ( 10th century A. D. ) who established it much before the Chinese and European scholars could do anything about it³ It is based on a concept which developed in India in association with the problem of making verses of different metres ( *chandas* ) in a poetical composition. The *Varṇasaṅgīta* (the music of sound variations) of the Vedic and post Vedic composers depended only on the variation of two sounds namely *guru* (long) and *laghu* (short) sounds. Piṅgala (200 B. C.) in his *Chandaḥsūtra*¹ described clearly a rule of finding the variation of sounds of syllables in a particular metre. For example, in a three syllabic *madhyā chanda* based on *guru* and *laghu*, the variations of *guru* and *laghu* will be — three *guru* sounds will occur once, two *guru* and one *laghu* sound will occur thrice, one *guru* and two *laghu* sounds occur thrice, three *laghu* sounds will occur once. If we take *guru* as *a* and *laghu* as *b*, then it gives (a+b)=1.a³+3a²b+3ab²+1.b³. Similarly for a four syllabic *Pratiṭhā chanda*, it uses (a+b)⁴=1.a⁴+4a³b+6a²b²+4ab³+1.b⁴. The technique of finding the number of variation of sounds was actually practised by Piṅgala and other Vedic composers² to detect the quality as well as shortcomings of the metres.

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² Vide Haran Chandra Banerjee's edition of *Līlāvatī*, p. 197-200.

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³ Bag, A. K. 'Binomial Theorem in Ancient India', *Indian Journal of History of Science*, 1, No. 1, p. 68-74, 1966.

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Halāyudha (10-11th century A.D.) in his commentary on the Piṅgala *Chandaḥsūtra*³ (ch. 8, verse 34) introduced a *meruprastāra* (pyramidal scheme) technique with the coefficients of the different variations in the expansion of (a+b)¹, (a+b)², (a+b)³………to minimise the trouble of computing the different layers which run thus :

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¹ *Chandaḥ sūtra* of Piṅgalācārya, edited with the commentary of Halāyudha by Sitanath Sarman, Calcutta.

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² Vide *Jayadaman*, edited by Velāṅkar.

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³ Ch. 8, Verse 34.

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“Draw a square on the top, two squares are drawn below (side by side) so that half of each is extended

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on either side. Below it three squares, below it (*again*) four squares are drawn and the process is repeated till the desired pyramid is attained. In the ( topmost ) first square the symbo' for one is to be marked. Then in each of the two squares of the second line figure one is to be marked. Then in the third line figure one is to be placed on each of the two extreme squares. In the middle square ( of the third line ) the sum of the figures in the two squares immediately above is to be placed ; this is the meaning of the term *pūrṇa*. In the fourth line one is to be placed in each of the two extreme squares. In each of the middle squares, the sum of the figures in the two squares immediately above, that is, three is placed. Subsequent squares are filled in this way. Thus the second line gives the expansion of combinations of ( short and long sounds forming ) in a one syllabic metre ; the third line the same for two syllables, the fourth line for three syllables *meruprastāra* and so on.” [ vide 1 Fig. 17 (a), 17 (b) and 17 (c). ]

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Fig. 17 (a).

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Fig. 17 (b).

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In modern notation this gives to :

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Fig. 17 (c).

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In this way, the general expansion of $(a+b)n = an + n_c₁ an-1 b + + bn was$ readily obtained for a metre of n syllables.

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The exactly same technique of arriving at the triangular array appeared in China (1303 A.D.) in the work *Ssu Yuan Yü Chien*¹ ('the Old Method Chart of Seven Multiplying Squares' ) of Chu Shih Chieh and in Europe in the publication of Pascal's *Traité du Triangle Arithmetic* (1665 A.D.) The Triangle, however, appeared in Europe more than hundred years before the *Traité* on the title page of the arithmetic of Apianus (1527 A.D.), and in the work of Stifel (1544 A D.), Scheubel (1545 A.D.), Tartaglia (1556 A.D.), Bombelli (1572 A.D.) and others¹, but about four centuries later before it actually appeared in India.

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¹ Mikamī, Y. *Mathematics in China and Japan*, New York, p. 90; Smith, D. E. *History of Mathematics*, 2, p. 508, Dover Publication; Needham, J. *Science and Civilisation in China*, 3, p. 134.

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¹ Smith, D. E. *History of Mathematics*, 2, p. 508-11, Dover Publication.

Linked texts (1)

Chandaḥsūtra — Piṅgala, Sanskrit