पृष्ठम्:Ganita Sara Sangraha - Sanskrit.djvu/३७९

181 by one (respectively), corresponding to the even (value) which is halved, and the unevoh (value from which one is subtracted, till by continuing these processes zero in ultimately reached. The numbers in the chain of figures so obtamed are all doubled, (and then in the process of continued multiplication from the bottom to the top of the chain, those figures which come to have a zero above them) are squared. The resulting prodact (of this continued multiplication) gives the number (of the varieties of stauzas possible in that syllabic metre or chaudas). CHAPTER VI--MIXED PROBI EMS. The arrangement (of short and long syllables in all the varie- ties of stanzas so obtained) is shown to be arrived at thus - (The natural numbers commencing with one and ending with the measure of the maximum number of possible stanzas in the given metre being noted down), every odd number (therein) has one ad to it, and is (then) halved. (Whenever this process is gone through), a long syllable is decidedly indicated. Where agaun odd, denotes a thd long syllable Thus the hist valety consists of thee long syllables and is indicated. thus . 2nd variety 2, being even, indicates a short syllable; when this 2 is divided by 2, the quotient is 1, which being odd indicates a long syllable. Add 1 to this 1, and divide the sum by 2, the quotient bemg odd indicates a long syllable thus we get | . Similarly the other six varieties are to be found out. (3) The fifth vanery, for mistance, may be found out as above. (+) To find out, for instance, the ordinal position of the variety | 2 | we proceed thus - Below these syllables, write down the terms of a series in geometrical pro- Add the gression, having 1 as the first term and 2 as the common iatio figures 1 and 1 under the the short syllables, and mcreare the sum ly 1, 12 4 we get & and we, therefore, say that this is the sixth variety in the t11-syllabic metre (5) Suppose the problem is How many varieties contain 2 short syllables > Write down the natural numbers in the regular and in the verse order, one 123 below the other thus Taking two terms fiom night to left, both from 321 above and from below, we divide the product of the former by the product of the latte. And the quotient 3 is the answer required (6) It is prescribed that the symbols representing the long and short syllables of any variety of metre should occupy an angula of vertical space, and that the intervening space between any two varieties should also be an angula The amount, therefore, of vertical space required for the S varieties of this metre is 2 S-1 or 15 angulas.