Mathematics in Ancient and Medieval India c7.s4.2

c7.s4.2

(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.