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Vaishali Institute Research Bulletin No. 4
Expansion of Varna Vrttas and the Binomial Expansion.
Let a denote a monosyllabic letters and b, a disyllabic one. All the variations of a metre having 1 letter are b and a; the variations of those having 2 letters are bb, ab, ba and aa (i.e., 2Cob3, 2C,ab and 2C,a2) and the variations of those having 3 letters are bbb, abb, bab, aab, bba, aba, baa and aaa (i.e., Cb, C1b2a, "C,ba2 and C,a) and so on. The result for a metre having n letters can be written as "Cob", "Cib"-la, "C2b"-2a2,..., "C,b"-"a",..., "Cbn, which is the binomial expansion of (b+a)n.
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Knowledge of certain properties associated with the above expansion such as the expansion itself, the total number of variations of the metre and the number of variations of a metre having a definite number of monosyllabic (or disyllabic) letters etc. were essential for the development of metric. The process for finding the above mentioned last property of an expansion has been named lagakriyā, galakriyā and laghukriya in metric. By this process the binomial conefficients and their relation with the terms of the expansion are obtained. Different rules for the formation of the binomial coefficients, arranged in their proper sequence, i.e., the (so called) Pascal's triangle, have been given by various writers. Thus, Varähamihira has given rule for the formation of the losta prastāra, Punāgacandra, that for the khande-meru and Halāyudha, that for the meru prastāra. Acārya Jayadeva has also given a rule for the formation of such a figure of number which is known as the suci prast ära in Sanskrit metric.
Suci Prastara.
Acarya Jayadeva's rule for the formation of the sūcī prastāra may be translated as given below.
"Keeping unity (at places) one more than the number of letters in the metre, vertically (one above the other and starting from the lowest), the 1st is added to the 2nd, the 2nd to the 3rd (and) after that, that to the 4th till the penultimate. (The process) is repeated again and again by leaving the addition to the penultimate (and) to that which becomes lower to it (in successive stages), in order."14
14.
Jd., viii, 8-10 (a). The text is, वृत्ताक्षराणि यावन्त्येकेनाधिकतराणि तावन्ति । ऊर्ध्वं क्रमेण रूपाण्यादौ विन्यस्य तेषां तु ॥ आक्षिपेद्वितीये द्वे च तृतीयेऽथ तान्यपि चतुर्थे । एवं यावदुपान्त्यं कुर्यात्वेवं हि भूयोऽपि ॥ ९ ॥ यदधो भवन्त्युपान्त्यात्तत्प्रभृति पुनः क्रमान्निवर्तन्ते ।
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