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109
The Binomial Expansion.
The rule for the binomial expansion as contained in the Ratna Mañjūsā and explained by its commentator may be stated as follows.
Vaishali Institute Research Bulletin No. 4
"After that, the numbers obtained are the numbers of variations of the metre having one monosyllabic letter, two monosyllabic letters etc. Keeping the numbers, separately, in order, the above ( rule ) should be stated in that order. "9
Thus, according to the rule, nCy, r = 0, 1, 2, ..., n are the numbers of variations of the metre having r monosyllabic letters, (r= 0, 1, 2, n), in order (or n-r disyllabic letters, r0, 1, 2, ..., n, in order), Correspondingly, nC, is the number of terms of the form a'bn' in the expansion of (a + b)* which is the binomial expansion of (a + b)n.
Merukriya
Merukriya is an advancement over lagakriya i. e., the binomial expansion. It is the process for finding the serial numbers of variations [each containing a definite number of monosyllabic (or disyllabic) letters] of a given metre amidst all variations of the metre. The Ratna Mañjüṣā contains a rule for such a merukriya based on the meru different from the meru as given by Halayudha and others.
Meru
Rule for the formation of the meru as contained in the Ratua Mañjūsā may be stated10 as given below.
9.
10.
Rm, viii, 17.
Rm., viii, 1-8 (p. 38-39). The text is, लगक्रियां सुधीः कुर्यात्तत्र येऽङ्काः स्थिताः क्रमात् । तत्तदङ्कप्रसंख्यातान् कोष्ठांस्तेषां पुरो लिखेत् ॥ ॥ सर्वगुर्वन्त सविधकोष्ठे तत्रैकमालिखेत् । तमेव द्विगुणीकृत्य तस्याधो द्वयङ्कमालिखेत् ||2|| as च द्विगुणीकृत्य चतुरङ्कमथा लिखेत् । तत्रैकमपनीयोर्ध्वपङ्क्तं त्रयङ्कमुपक्षिपेत् ।। 34त्र्यङ्क च द्विगुणीकृत्य षडङ्क तदधः क्षिपेत । तत्रैकमपनीयोर्ध्वपङ्क्तौ पंच लिखेत्ततः ॥4॥ The author continues, चतुरङ्कद्विगुणनात् अष्टौ च तदधः क्षिपेत् । तत्रैकापनयात् सप्ताप्यूर्ध्वपङ्क्तौ निवेशयेत् ॥15॥ एवं द्विगुणितान् पंचादिकानन्तानधः क्षिपेत् । तानवैकोन कानूर्ध्वङ्क्तौ कुर्वन्नमुं विधिम् ||6||
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