A history of Pingala's combinatorics (Jayant Shah)

8/9/2019 A history of Pingala's combinatorics (Jayant Shah)

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A HISTORY OF PI ! GALA’S COMBINATORICS

JAYANT SHAH

Northeastern University, Boston, Mass

Abstract: Combinatorics forms an important chapter in the history of Indian mathematics. The

tradition began with the formal theory of Sanskrit meters formulated by Pi! gala in the 2nd

century B.C.E. His recursive algorithms are the first example of recursion in Indian

mathematics. Pi! gala’s calculation of the binomial coefficients, use of repeated partial sums of

sequences and the formula for summing a geometric series became an integral part of Indian

mathematics. This paper systematically traces the extent to which Pi! gala’s algorithms have

been preserved, modified, adapted or superseded over the course of one and a half millennia. It

also addresses Albrecht Weber’s criticism about Hal " yudha’s attribution of the construction ofwhat is now known as Pascal’s triangle to Pi! gala. While agreeing with Weber’s criticism of

Hal " yudha, this paper also faults Weber’s interpretation of Pi! gala, but shows that the

construction can still be traced to Pi! gala.

!" $%&'()*+&$(%

After giving an exhaustive account of Sanskrit meters in Chanda#$" stra in the 2nd centuryB.C.E. , Pi! gala concludes with a formal theory of meters. He gives procedures for listing all possible forms of an n-syllable meter and for indexing such a list. He also provides an algorithmfor determining how many of these forms have a specified number of short syllables, that is, an

algorithm for calculating the binomial coefficients nCk . What is striking is that this is a purelymathematical theory, apparently mathematics for its own sake. Unlike P "%ini’s Sanskritgrammar, Pi! gala’s formal theory has no practical use in prosody. However, his mathematicalinnovations resonate with modern mathematics. Pi! gala consistently uses recursion in hisalgorithms, a tradition which stretches from P "%ini (6th century B.C.E.) to & ryabhata (5th centuryC.E.) to M "dhava (14th century). & ryabhata provides a sine table which is the same as the tableof chords of Hipparchus. Both tables rely on triogonometric identities for their construction. & ryabhata then proceeds to give a recursive algorithm for generating the same table. Thealgorithm is essentially a numerical procedure for solving the second order differential equationfor sine and thus necessarily less accurate than the one based on triogonometric identities.

M "dhava follows & ryabhata and develops the power series for sine, cosine and arctangent usingrecursion.

A remarkable example of the mathematical spirit of Pi! gala’s work is his computation of the powers of 2. He provides an efficient recursive algorithm based on what computer scientists nowcall the divide-and-conquer strategy. Another example is his formula for the sum of the

geometric series with common ratio equal to 2. It was generalized to an arbitrary common ratio by ' ridhara (c. 750 C.E.). Curiously, following Pi! gala, the formula for the geometric series is


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almost always combined with Pi! gala’s divide-and-conquer algorithm. Even in the 14 th century, N "r " ya%a in Ga%itakaumudi repeats Pi! gala’s algorithm almost verbatim for summinggeometric series. ' ridhara also provided the modern day formula for calculating the binomialcoefficients which replaced Pi! gala’s recursive algorithm in Indian mathematics. The prosodistsstill continued to use Pi! gala’s method. Pi! gala’s algorithms were generalized by S "r ! gadeva to

rhythms which use four kinds of beats - druta, laghu, guru and pluta of durations 1, 2, 4 and 6repectively (Sa! g ( taratn"kara, c. 1225 C.E.). In another direction, N "r " ya%a developed manydifferent series and their applications in Ga%itakaumudi (see Kusuba1).

There is a mystery surrounding Pi! gala’s computation of nCk . It is almost universally acceptedon the authority of Hal " yudha (10th century, C.E.) that Pi! gala’s last s)tra, “ pare p)r !a*iti”,implies the construction of meru prast "ra (what is now known as “Pascal’s triangle”) forcomputing nCk . However, except for Virah"!ka, none of the authors (from Bharata – 1

st century

C.E. onwards) before Hal " yudha describes such a construction or even employs the designationmeru prast "ra for the construction they do describe. This is strange in view of the fact that thealgorithms of Pi! gala have been copied and elaborated by later authors for more than a

millennium after Pi! gala. Albrecht Weber

2

in 1835 commenting on Hal " yudha’s interpretationflatly declares that “that our author ( Pi! gala) may have had in mind something like meru prast "ra does not follow from his words in any way”. Alsdorf 3 in 1933 asserts that Weber’sstatement has no foundation and that Weber misunderstood Hal " yudha. He then goes on toconjecture, without presenting any evidence, that the repetition of the s)tra “ pare p)r !a*” at theend of the composition is a later addition and that it was inserted as a reference to meru prast "ra invented later. Authors from Bharata onwards do describe a construction which yields the sametriangle. Instead of filling the triangle from the top, they do so diagonally from left to right.

Although both constructions yield the same triangle, they are algorithmically quite different andin fact, Virah"!ka includes both with evocative names s)ci (“needle”) prast "ra and meru (“mountain”) prast "ra. The method described in Bharata’s N "tya $" stra is based on a techniquevery common among Indian mathematicians, namely, creating a new sequence from a given

sequence by listing its partial sums. Indeed, summing sequences of partial sums became astandard topic in Indian mathematics. Aryabha+ a provides a formula for the sum of sequences of partial sums of the sequence 1,2,3, !!!,n which is the binomial coefficient n+2C3. The sum

1+2+!!!+n is the binomial coefficient n+1C2. N "r " ya%a in the 14th

century gave a general formula

in the form of binomial coefficients for summing sequences obtained by repeatedly forming

partial sums an arbitrary number of times. Sequences of partial sums were crucial also in M "dhava’s development of infinite series. N "tya $" stra evidently relies on the work of the past

# “ Combinatorics and Magic Squares in India, A Study of N!r !ya"a Pa"#ita’s Ga%itakaumudi,

Chapters 13-14”, Takanori Kusuba, Ph.D. Thesis, Brown University, May, 1993. " “Ueber die Metrik der Inder”, Albrecht Weber, Berlin, (1863). Translation of quotes from

Weber provided by Gudrun Eisenlohr and Dieter Eisenlohr.

$“Die Pratyayas. Ein Beitrag zur indischen Mathematik”, Ludwig Alsdorf, Zeitschrift für

Indologie und Iranistik, 9, (1933), pp. 97-157. Translated into English by S.R. Sarma, Indian

Journal of History of Science, 26(1), (1991)


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masters and the construction must go back to Pi! gala. None of the prosodists following Pi! gala acknowledges Bharata, but they do acknowledge Pi! gala.

This paper systematically traces Pi! gala’s algorithms through the Indian mathematical literatureover the course of one and a half millennia. It finds no evidence to support Hal " yudha’s

interpretation of Pi! gala’s last s)tra, but still traces the computation of the binomial coefficientsto Pi! gala. Especially relevant are the compositions of Bharata and Jan"$raya which arechronologically closest to Pi! gala. The section on Sanskrit meters in Bharata’s N "+ ya $" stra (composed sometime between 2nd century BCE and 1st century CE) still has not been translated.Words are corrupted here and there and some of verses appear out of order. Regnaud

4 in his

monograph on Bharata’s exposition on prosody concedes that a literal translation is not possibleand skips many verses without attempting even a loose interpretation. In his 1933 paper on

combinatorics in Hemacandra’s Chandonu $asanam, Alsdorf establishes a loose correspondence between Bharata and Hemacandra without translating N "+ ya $" stra. The Sanskrit commentary of Abhinavagupta (c. 1000 CE) on N "+ ya $" stra is spotty and frequently substitutes equivalentalgorithms from later sources instead of explaining the actual verse. J "n"$rayi of Jan"$raya (c.

6

th

century CE) is absent from the literature on Pi! gala’s combinatorics. Even in his otherwiseexcellent summary of Indian combinatorics before N "r " ya%a, Kusuba barely mentions Bharata and does not mention Jan"$raya. In this paper, we give translations of both works. Even in places where the literal text is unclear, its mathematical content is unambiguous. We also give

translations of V , ttaj"tisamuccaya of Virah"!ka, Jayadevacchanda# of Jayadeva,Chandonu $asana* of Jayak ( rti and V , ttaratn"kara of Ked "ra which have not yet been translatedinto a western language. In the case of V , ttaj"tisamuccaya, which was composed by Virah"!ka in Pr "k , ta, only its Sanskrit version rendered by his commentator is given. The survey in this paper is based on the following primary sources:

Date Title Author Translation

c. 2

nd

century BCE Chanda#$" stra Pi! gala Included2nd century BCE to 1st century CE N "+ ya $" stra Bharata Includedc. 550 CE B, hatsa*hita Var "hamihira Refer to Kusubac. 600 CE J "n"$ray( Chandoviciti# Jan"$raya Included

c. 7th

century CE V , ttaj"tisamuccaya Virah"!ka Includedc. 750 CE P "+ iga%ita ' ridhara Refer to Shuklac. 850 CE Ga%itasarasa! graha Mah"vira Included

before 900 CE Jayadevacchanda# Jayadeva Includedc. 950 CE M , tasañj( van( Hal " yudha Refer to Weberc. 1000 CE Chandonu $asana* Jayak ( rti Includedc. 1100 CE V , ttaratn"kara Ked "ra Included

c. 1150 CE Chandonu $asanam Hemacandra Included1356 CE Ga%itakaumudi N "r " ya%a Refer to KusubaUnknown Ratnamañj)-a Unknown Refer to Kusuba

% “La Métrique de Bharata”, Paul Regnaud, Extrait des annals du musée guimet, v. 2, Paris,

(1880).


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The list does not include Svaya*bh)’s Svaya*bh)chanda# and Chanda $cityuttar "dhyaya (20th chapter) of Br "hmaspu+ asiddh"nta (628 CE) of Brahmagupta. The former does not covercombinatorics. Tantalizingly, Brahmagupta’s text does contain prosodist’s technical terms like

na -+ a*, uddi -+ a* and meru, but the text is too corrupted to make any sense out of it; even the

commentator P , thudaka skips over it. The list also omits the pur "%as because their description isessentially identical to descriptions in the sources listed above. Descriptions in Agnipur "%a andGarudapur "%a are identical and closely follow Pi! gala’s s)tras except that they replace Pi! gala’s prast "ra (method of listing meters) with a version given in N "+ ya $" stra. They alsoassert that the “lagakriy"” (algorithm for computing nCk ) is executed by means of the “meru- prast "ra” without actually describing the algorithm. The description in N "radapur "%a is almostthe same as the one in Ked "ra’s V , ttaratn"kara.

Pi! gala’s algorithms and their modifications by later authors are described in the next section.Translations of the relevant sections of the texts listed above are given in Section 3 to provide adetailed chronological history.

," PI ! GALA’S ALGORITHMS In the following, G denotes a long (Guru) syllable while L denotes a short (Laghu) syllable.

There are two classes of meters in Sanskrit:

1. Ak -arachanda#: These are specified by the number of syllables they contain. The vedicak -arachanda#, referred to as Chanda#, are specified simply by the number of syllables.The later ak -arachanda#, called V , ttachanda#, consist of 4 feet ( p"da), each foot havinga specified sequence of long and short syllables.

2.

those which are measured by the number of m"tr "s they contain. A m"tr " is essentially atime measure. A short syllable is assigned one m"tr " while a long syllable is assignedtwo. There are two kinds of these meters.

(a) Ga%achanda#: meters in which the number of m"tr "s in each foot ( ga%a) isspecified.

(b) M "tr "chanda#: meters in which only the total number of m"tr "s is specified.

Pi! gala’s algorithms deal only with the V , ttachanda#. These are of three types. Sama (equal) arethe forms in which all four feet have the same sequence of short and long syllables. In

ardhasama (half-equal), the arrangement of short and long syllables in the odd feet is differentfrom that in the even feet, but each pair has the same arrangement. The forms which are neither

sama or ardhasama are called vi -ama (unequal). Although Pi! gala does list ga%achanda# andm"tr "chanda# that were employed in prosody at that time, much of their development came at alater time at the hands of Pr "k , ta prosodists. Each prosodist after Pi! gala has something to sayabout the combinatorics of ga%achanda# and m"tr "chanda#.

Sanskrit prosodists traditionally identify the following six formal problems and their solutions.

(called pratyayas). ( Pi! gala does not assign any labels to them.)


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&-. /$0 !"#$%#%#/

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Prast "ra '()*+, Systematically lists all theoretically possible forms of a meter

with a fixed number of syllables. Na -+ a* -../0/1+2*,3 4562 Recovers the form of a meter when its serial number in the list is

given.

Uddi -+ a* 7.,/8+2*, Determines the serial number of a given form. Lagakriy" '05)2945.:9

*;*)8/6*

Calculates the number of forms with a specified number of short

syllables (or long syllables).

Sa!khy"


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LLG

GGL

LGL

GLL

LLL

Prosodists later generalized these algorithms to list forms of m"tr "-meters.

2.2 Na#$ a% The next algorithm determines the index (serial number) of a given form in the list. Notice thatforms with an even index begin with L while those with an odd index begin with G. If we

remove the first column, what remains is a list obtained by writing every form of the (n-1)-syllable meter twice. Therefore, index of the string of letters remaining after removing the first

letter of a form is half of the original index if it is even and half of original index increased by 1if it is odd. To write down the form corresponding to a given index k , write L as the first letter if

k is even, G otherwise. Divide k by 2 if it is even, add one to it and divide by 2 if k is odd. This isnow the index of the remaining string. Repeat the process until you have written down all the n

letters. For example, if k = 66 is even, put L and halve it

3 is odd, add 1 before halving, put G2 is even, put L, halve it.

1 is odd, add 1 before halving, put GWhen you reach one, the algorithm produces a series of Gs and is continued until the requisite

number of syllables has been obtained. Thus, for meters of length 3, 4 and 5, you get LGL,

LGLG and LGLGG respectively in the sixth position.

2.3 Uddi #$ a% Uddi -+ a* is used to find the index of a given form. !" ! gala’s algorithm simply reverses the

process used in na -+ a*. Start with the initial value equal to 1. Scan the given form from right toleft. At the first L, double the initial 1. From then on, successively double the number wheneveran L is encountered. If a G is encountered, subtract one after doubling.

Example: LGLGInitially, k=1.

Start at the last L: LGLG, get k = 2 The next letter is G: LGLG, Get k = 2x2-1=3.

The next letter is L: LGLG, Get k = 2x3=6 .

Mathematician Mah"v( ra replaced this algorithm by a method which amounts to aninterpretation of the string as a binary number. Write down the geometric series starting with one

with common ratio equal to 2 above the syllables of the form. (Effectively, he wrote down aboveeach letter its positional value in terms of the powers of 2.) The index is then one plus the sum of

the numbers above all Ls in the string. This is the closest Indian mathematicians came toinventing binary numbers. Later prosodists with the exception of Hemacandra follow Mah"v( ra.This method was probably known as far back as the 1

st century. Bharata’s Verse (121) given in

the Section 3.3.2 seems to outline this method.


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2.4 Lagakriy" Lagakriy", answers the question: Out of all the forms of an n-syllable meter, how many haveexactly k syllables? In modern notation: how to calculate the binomial coefficient nCk ? Allmodern authors seem to have accepted Hal " yudha’s attribution of the original algorithm to!" ! gala, but it is hard to see any connection between the words of the s)tra cited by Hal " yudha

and his interpretation that it is !" ! gala’s algorithm for calculating nCk . The s)tra by itself has noinformation for carrying out the construction described by Hal " yudha. One of the mainobjectives of this paper is to settle this mystery by tracing the history of this algorithm throughthe available works of all the prosodists before Hal " yudha. It will be shown that the originalalgorithm can be traced to !" ! gala, but not to the s)tra cited by Hal " yudha, but another s)tra which he seems to omit or is unaware of. Hal " yudha claims that the last s)tra of !" ! gala refersto meru-prast "ra (Pascal’s triangle). Here are !" ! gala’s last two s)tras:

pare p)r !a* “next, full” pare p)r !a*iti “next, full, and so on”

The two s)tras are identical except that the second has iti (“and so on”) appended to it. Hal " yudha correctly interprets the first s)tra. It is the formula: twice the number of forms of ameter equals the number of forms of the next meter. !" ! gala writes “full” instead of “twice” because the formula in the previous s)tra subtracts 2 from the doubled number. The word p)r !a* tells us to restore the diminished double to its original full value. Even though the last s)tra is identical, Hal " yudha inteprets it as an instruction to construct the meru (a mythicalmountain) for calculating the binomial coefficients. He then gives detailed instruction for the

construction:

“First write one square cell at the top. Below it, write two cells, extending half way on bothsides. Below that three, below that again, four, until desired number of places (are obtained.)

Begin by writing 1 in the first cell. In the other cells, put down the sum of the numbers of the twocells above it. !!!”

Thus, for n=6, we get the following table:

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

The numbers in the bottom row are the number of forms with 0, 1, 2, 3, 4, 5 and 6 Lsrespectively.

Weber who totally rejects Hal " yudha’s attribution translates the two s)tras together as:

“The following (meter comprises) the full (twice the sum of the combinations of the previous

meter without subtracting 2).”


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It is hard to argue against Weber’s interpretation which is what the actual words are saying or to

find evidence to support Alsdorf’s argument in defense of Hal " yudha. It is very likely that Hal " yudha based his claim on an earlier source, but the only reference I could find is a mentionin Agnipur "%a and Garudapur "%a which have the following half-verse:

pare p)r !a* pare p)r !a* meruprast "ro bhavet |

“next full, next full, meru prast "ra is created”

These pur "%as are the only place where !" ! gala’s phrase, pare p)r !a* pare p)r !a*, appearsagain. Since pur "%as are compiled perhaps by several authors over centuries, Hal " yudha’ssource probably goes quite far back, but seems lost. Still, I don’t see any way to justify

Hal " yudha’s interpretation after reading all the available sources. The most reasonableinterpretation of the repetition of the phrase pare p)r !a* is that it is the standard Sanskrit usagefor indicating a repeated action. So the two s)tras together just mean the recursive formula S n+1 =

2S n where S n is the total number of forms of the n-syllable meter.

I think the computation of binomial coefficients can still be traced back to !" ! gala. There is a s)tra quoted by Weber from the Yajur recension of !" ! gala’s Chanda#$" stra which does exactlythat. Hal " yudha seems to be unaware of this, for it does not appear in his text. The s)tra inquestion is the following:

ekottarakrama $a# | p)rvap, kt " lasa*khy" || (23b)

Weber couldn’t get any coherent sense out of this. He translates the s)tra as follows:

“Step-by-step always by adding one, the number of la is always combined with the previous(number).”

He then comments: “The ‘number of la (referring to the last word lasa*khy" in the s)tra )’ canonly mean the number of combinations, which for each of the following meter is twice that of the preceeding. This meaning of ‘la’ is no more provable than the required meaning (of ‘la’) in Rule

22 ( s)tra 8.22) as ‘syllable’. Also the phrase p)rvap, kt " (in the s)tra) is not suitable to designatethis doubling. Furthemore, the description of this doubling is expressly found in Rule 33 ( s)tra 8.33) below. This mention here is strange. For the time being, I cannot see another explanationfor the Rule 23b than the one given above. It is interesting to compare Kedara’s (6.2-3)

description with this. !!!.” Weber then goes on to describe the prast "ra algorithm given in

Ked "ra’s V , ttaratn"kara (verses 6.2, 6.3) which makes even less sense. Van Nootan5

acceptsthis suggestion and offers Kedara’s method of enumeration as a clearer version of !" ! gala’s s)tra.

& “Binary Numbers in Indian Antiquity”, B. van Nooten, Journal of Indian Philosophy, 21, pp.

31-50, (1993).


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I will try to show that this cryptic s)tra actually describes lagakriy". It literally translates asfollows:

ekottara = “increasing by one”. krama $a# = “successively, step by step, sequentially”. p)rvap, kt " = “mixed with the next”. lasa*khy" = “L-count”.

In order to decipher this, consider a more detailed version in N "+ ya $" stra:

ek "dhik "* tath" sa*khy"* chandaso vinive $ ya tu | y"vat p)rñantu p)rveña p)rayeduttara* gaña* || (124)eva* k , tv" tu sarve -"* pare -"* p)rvap)raña* |kram"nnaidhanamekaika* pratiloma* visarjayet || (126) sarve -"* chandas"meva* laghvak -aravini $caya* | j"n( ta samav, att "n"* sa*khy"* sa*k -epatastath" || (127)

“Put down (a sequence, repeatedly) increased by one upto to the number (of syllables) of the

meter.Also, add the next number to the previous sum until finished.

Also, after thus doing (the process of) addition of the next, (that is, formation of partial sums) ofall the further (sequences),

Remove one by one, in reverse order, the terminal (number) successively.Of all meters with (pre)determined (number of) short syllables

Thus know concisely the number of sama forms”

What is striking is the close correspondence between the key words in Pi! gala 's s)tra and the phrases in Bharata's verses. Pi! gala 's s)tra begins with the word ekottara while Bharata beginswith the word ek "dhika*6 . Both terms are used by medieval mathematicians to indicatearithmetic series with common difference equal to one (eka). The rest of the first line in

Bharata's verses specifies the length of the sequence which Pi! gala does not mention explicitly.

Pi! gala 's compound p)rvap, kt " parallels Bharata's phrase, p)rvena p)rayet in the second lineand more closely, to its compound version, p)rvap)raña* in the third line. Pi! gala and Bharata both employ the term p)rva. It has a multitude of meanings, but what makes sense in the presentcontext is the interpretation "in front or next". p, kt " is the past passive participle of p, c whichmeans "to mix or to join". It modifies the noun lasa*khy". Bharata employs the verb p. (to fill,to complete, to make full) and uses its derivatives p)rayet and p)raña*. p)rayet is the causative(optative mode) of p. while p)raña* is a derived noun meaning the act or the process of filling.In the context of arithmetic, both p, c and p. may be interpreted to mean “to add”.

Bharata's second line explains the process of p)rvap)raña*: “ y"vat (until ) p)rña* (completed) tu (also) p)rveña (with the next) p)rayet (fill) uttara* (the previous) gaña* (sum).” The thirdline of his algorithm then employs p)rvap)raña* repeatedly to construct sequences of partial

? The texts of Regnaud and Nagar have the word ek "dik "* (numbers beginning with one) instead

of ek "dhik "* (numbers increasing by one) which is what Alsdorf has.


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sums: “eva* (thus) k , tv" (after doing) tu (also) sarve -"* (of all, every) pare -"* (further, beyond) p)rvap)raña* (the process of adding the next).

Pi! gala uses p)rvap, kt " in place of p)rvap)raña* and omits the detail of Bharata's second line.He also employs krama $ah (step by step) instead of Bharata's phrase sarve -"* pare -"* to

indicate a repeated action. With these identifications, Pi! gala 's krama $ah p)rvap, kt " becomesequivalent to Bharata's third line. It tells us to repeatedly calculate sequences of partial sums.

Clearly, Pi! gala 's designation lasa*khy" must refer to the number of forms with the specifiednumber of Ls, that is, Bharata's last two lines. The whole s)tra may now be interpreted assaying: “The number of forms with the specified number of Ls is obtained by the process ofrepeatedly calculating the partial sums of the intial sequence 1,2, !!!,n.” The detail about

systematically stripping the terminal partial sums given in Bharata's fourth line is missing.

Following Bharata's recipe, with n = 6 , we get the following table:

65 154 10 20

3 6 10 152 3 4 5 6

1 1 1 1 1 1

Read from bottom to top, each column consists of partial sums of the sequence in the previouscolumn. The last partial sum in each case is omitted. The numbers 6, 15, 20, 15, 6 and 1 along

the diagonal are numbers of forms with 1, 2, 3, 4, 5 and 6 Ls respectively.

The recursive procedure rests on the fact that the forms with k short syllables of an n-syllablemeter may be obtained by appending L to the forms with (k-1) short syllables of the (n-1)-syllable meter and appending G to its forms with k short syllables. That is, nCk = n-1Ck + n-1Ck-1 if

k


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Mathematicians soon adopted the combinatorial problem of choosing k objects out of n as a

standard topic in Indian mathematics and illustrated it with a wide variety of examples.Var "hamihira ( B, hatsa*hit ", Adhy" ya 76, Verse 22) extended the procedure and outlined amethod for listing the actual nCk combinations. Ratnamañj)-a quotes an algorithm by an

unnamed author for listing the serial numbers of nCk combinations. Beginning with ' ridhara inthe 7th century, mathematicians started using the modern fomula for calculating nCk :

nC k =

n(n "1)(n " 2)# ## (n " k +1)

1# 2# 3## # k

2.5 Sa&khy" Pi! gala uses recursion also in his fifth algorithm which calculates the number S n of all possibleforms of a meter. He notes that the number of forms of the n-syllable meter is twice the numberof forms of the (n-1)-syllable meter. One can calculate S n simply by repeated doubling, but

Pi! gala gives a recursive method for calculating 2n

which rests on the fact that if n is even,2n=(2n/2)2. So the recursion is: if n is even, 2n=(2n/2)2 and if n is odd, 2n=2!2(n-1). Pi! gala sets it upas follows:

dvirardhe| r ) pe $)nyam| dvi# $)nye| t "vadardhe tadgu%itam|“two in case of half. (If n can be halved, write ‘twice’.)

In case one (must be subtracted in order to halve), (write) ‘zero’.(going in reverse order), twice if ‘zero’.

In case where the number can be halved, multiply by itself (that is, square the result.)”

For example, with n = 6:

Construct the first two columns in the table below and then going back up, construct the third:n=6 twice (2!22)2 = 64

n=3 zero 2!22 = 8

n=2 twice 22 = 4

n=1 zero 2

Thus, total number S 6 of possible forms of length 6 is 64.

The prosodists later added another method: S n is equal to the sum of all the numbers obtained bylagakriy". More importantly, Virah"!ka extended the recursion to the computation of thenumber of forms of m"tr "-meters: S n= S n-1+ S n-2 which of course generates what is now knownas the Fibonacci sequence.

Pi! gala abstracts the fact that the list of forms of an n-syllable meter contains lists of forms of allthe meters with fewer syllables and presents it as the sum of a geometric series:

dvirdv)na* tadant "n"m|

“twice two-less that (quantity) replaces (the sequence of counts) ending (with the currentcount).”


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That is, 2S n – 2 = S 1 + S 2 + !!! + S n. More explicitly, 2!2n – 2 = 2

1 + 2

2 + !!! +2

n, a formula for

summing a geometric series.

Mathematicians from ' ridhara on incorporated geometric series as an integral part of

mathematics. Their formula is always coupled with Pi! gala’s recursive algorithm for computing powers, attesting to the lasting influence of Pi! gala on Indian mathematics. For example, here isa quote from ' ridhara in the 7th century:

vi -ame pade nireke gu%a* same’rdh( k , te k , ti* nyasya |krama $o rupasyotkrama $o gu%ak , tiphalam"din" gu%ayet || (94) pr " gvatphalam"dy)na* nirekagu%abh" jita* bhaved ga%ita*| (95-i)

“When the number of terms (of the series) is odd, subtract 1 from it and write ‘multiply (by thecommon ratio)’. When even, write ‘square’ after halving it. (Continue this) step by step (until the

number is reduced to zero). Starting with 1, step by step in reverse order, multiply (by the

common ratio) and square the number (as the case may be), multiply the final result by the firstterm. (The result is the next term in the series.) The result obtained as above, diminished by thefirst term of the series and then divided by the common ratio diminished by one is the sum.”

Thus, we get the fomula

a+ ar + ar 2+" " " + ar

n

=

ar n+1

# a

r #1

N "r " ya%a in the 14th century has the same formulation:

vi -ame pade vir ) pe gu%a# same’rdh( k , te k , ti $c"nty"t|

gu%avargaphala* vyeka* vyekagu%" pta* mukh"h, ta* ga%ita*||

“When the number of terms (of the series) is odd, subtract 1 from it and write ‘multiply’ (by the

common ratio). When even, write ‘square’ after halving it. The result obtained by multiplyingand squaring in reverse order starting from the last, diminished by one, divided by the common

ratio diminished by one and (finally) multiplied by the first term is the sum.”

2.6 Adhvayoga

The last algorithm is a strange computation of the space required to write down all the forms of a

particular meter. Allowing for a space between successive forms, the space required is twice thenumber of forms minus one. Jan!$raya sets the width of each line equal to the width of a finger

and calculates that the space required to write down forms of the 24-syllable meter is 33,554,431finger-widths or about 265 miles. Clearly not a practical proposition! One has to assume that the

calculation is meant to demonstrate the immensity of the list.


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?" &'@%/4@&$(%/

In the following, the translations appear in quotes. Comments in parantheses are added to clarify

the meaning. The numbering of the verses is the same as their numbering in the primary sources

listed at the end of this section.

?"! !"#'$("#

)*+*+ !, & gala

dvikau glau | (8.20) “Double GL-pair.”

We get the following:G

L

G

L

mi $rau ca | (8.21) “(the pair of GLs) mixed and.”

Mixing GL-pair with another GL-pair results in GGLL. This is then placed in the secondcolumn. The result is:

GG

LG

GL

LL

p, thag (g?)l " mi $r "# | (8.22) “Repeatedly GLs mixed.”

GGG

LGG

GLG

LLG

GGL

LGL

GLL

LLL

vasavastrik "#

| (8.23) “eight triples.”

These are the 8 triples listed above. Pi! gala names (codes) these triples as m, y, r, s, t, j, bh andn respectively.

Pi! gala does not give a procedure for enumerating forms specified by fixing the number ofm"tr "s, that is, the m"tr "chanda#, but merely lists the five possible forms of 4-m"tr " feet,


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#&

n=3LG

GL

LLL

n=4

GGLLG

LGL

GLL

LLLL

and so on. Notice that the list for n=4 is exactly the one given by Pi! gala.

The verses (114) and (115) below describe the first version of Bharata’s prast "ra while the verse(116) describes the second.

guroradhast "d "dyasya prast "re laghu vinyaset |agratastu sam"dey" gurava# p,-+ htastath" || (114) prathama* gurubhirvar %airlaghubhistvavas"najam |v, ttantu sarvachandassu prast "ravidhireva tu || (115)

“Below the first G in the prast "ra, put down LFurther (i.e. after L) also, the same (as the combination above) to be bestowed, but Gs behind(i.e. before L) thus.”

“The first form (is filled) with syllable Gs, but the last with Ls.

Thus (is) the prast "ra procedure in the case of all meters.”

The enumeration is begun by writing down n Gs in the first line. Subsequently, to write down thenext combination, write L below the first G in the line above and copy the rest of the line after

its first G. Fill the space before the L with Gs. For example,GGG

LGG

GLG

LLG

GGL

LGL

GLL

LLL

An alternate version is:

gurvadhast "llaghu* nyasya tath" dvidvi yathoditam |nyasyet prast "ram"rgo’yamak -aroktastu nitya $a# || (116)

“G below L to be put down thus repeatedly doubled as has been said


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put down this course of prast "ra (when) syllable-specified, always.”

In the first column, write down Gs alternating with Ls. In the second column, write two Gsalternating with two Ls. In the third column, write four Gs alternating with four Ls. Continue to

fill the successive columns by double the number of Gs alternating with double the number of Ls

. This produces the following:

GGGG . . .

LGGG . . .

GLGG . . .

LLGG . . .

GGLG . . .

LGLG . . .

GLLG . . .

LLLG . . .

.

..

Once the required number of syllables is reached, it is clear how many rows are to be retained.

For example, for meters of length of two, only two columns are needed and only four rows needto be retained since subsequent rows are copies of the first four.

m"tr " sa*khy"vinirdi -+ o ga%o m"tr "vikalpita# |mi $rau gl "viti vijñeyau p, thak lak - yavibh" gata# || (117)

“A foot ( ga%a) specified by the number of m"tr "s is a m!tr !-combinationKnowing ‘mixed GL’, (apply) repeatedly according to intended apportionment.”

This verse may be just to reinforce what is indicated in verse (113). Alternately, it suggests that

the first method may be used for m"tr "-meters as well provided that syllables are adjusted to getthe correct number of m"tr "s. This is how the later prosodists adjust this algorithm to list formsof a m"tr "-meter.

m"tr " ga%o guru $caiva laghun( caiva lak -ite |"ry"%"* tu caturm"tr " prast "ra# parikalpita# || (118)

“(Forms of) a matr "-foot (consiting of) G and two Ls, having been attended to,also (obtain) (the meter) & ry"’s four-matra-prast "ra by combinations.”

The meaning is clear. & ry" uses feet consisting of four m"tr "s.

g ( takaprabh, t ( n"ntu pañcam"tro ga%a# sm, ta# |vait "l ( ya* purask , tya -a%m"tr "dy" stathaiva ca || (119)

(meters) G( taka’s and Prabh, t ( ’s five-m"tr " feet also, (as) recounted,


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(meter) Vait "l ( ya placed in front (i.e. beginning with Vait "l ( ya), (feet classes) beginning with 6-m"tr " (feet) similarly, and.”

Here, clearly an open-ended list of feet of increasing lengths is indicated.

tryak -ar " stu trik " jñey" laghugurvak -ar "nvit "# |m"tr " ga%avibh" gastu gurulaghvak -ar "$raya# || (120)

“Three-syllable (feet), triples, known by L-G syllable combination. M "tr " feet classes also syllables G-L dependent.”

)*+*) 304561070

( Jan" sraya denotes long syllables as ‘bha’ and short syllables as ‘ha’. After stating each s%tra, Jan"$raya also provides a commentary.)

prast "ra# (6.1) “ Prast "ra”

"dye bh" (6.2) “At the beginning, Gs.>

sarve -"* chandas"m"dye v, tte gurava eva bhavanti | etadukta* bhavati | sarve -"* chandas"m"ñca v, tta* sarvagurviti |

“Only Gs occur in the first form (in the listing) of all meters. This is what is said. All Gs in thefirst form of all meters. (GGGG…)”

"dau h" (6.3) “In the first place, L.”

sarvaguruv, ttam"dya* k , tv" tato dvit ( yasya v, ttasya chedane dvit ( yav, tt "dau laghureka# k "ryaiti |

“After creating all-G first form, then, dividing the second form (into two parts), place one L atthe beginning of the second form.”

p)rvavacche -a* (6.4) “The remainder (the rest) as before.”

dvit ( yav, tt "dau laghumeka* k , tv" tato’nantaramupari nyasta* p)rvav, ttamiva $e -a* kury"t | $e -amiti kriyam"%av, tta $e /a* | ida* prast "ranetradvis)trayuk sarve -"* chandas"* yuk

v, ttaprast "re yojya* |

“After placing one L at the beginning of the second form, then, next (to L), the remainder is

made up just like the previous form placed above. ‘remainder’ means making up the remainder(after the first L) of the form. Such a two-piece union as prast "ra lead (leading piece, LGGG…)must be employed in even (numbered) forms of the list in the case of all meters.”

bh"hatuly"# (6.5) “GL to be balanced.”


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bhau bhau pare (6.9) “two Gs, two Gs, next.”

te -"* tatha nyast "n"* parato dvau gurulagh) c"dho’dha# k "ryau | p)rvamek "ntaritanyasta- gurulaghusa*khy" pram"%apr " pteriti tadvipar ( ta ityanuvartate | tatha nyast "n"* te -"* paratodvigu%a* dvigu%a* guravo laghava $c"dho’dha# k "ry"# | "chandok -arapram"%aparisam" pte-

rityuktamevartha* nir ) payi - y"ma# | "dy" dvigu%akriya catv"ro gurava $ca laghava $catato’ -+ h"veva dvigu%advigu%av, ddhi* sarve -"* chandas"m"cchandok -araparisam" pteriti |

“After putting down those (referring to 6.8), put down next pairs of Gs and pairs of Ls,repeatedly one below the other. (In the next sentence, the exact meaning of tadvipar & ta is unclear,

but it says something like:) Having obtained the total number from the alternating GLs put downearlier, (the new column) follows the previous form modified. Having put these down thus, put

down repeatedly doubled Gs and Ls, one below the other (in successive columns.) This procedure is completed when the number of syllables is reached, (the number of columns equals

the number of syllables); we wish to indicate that this is the meaning of what is said. Doubling inthe beginning, four Gs and four Ls, and then, eight of the two, repeatedly increased by doubling,

completed when the number of syllables is reached, (thus) for all meters.”

)*+*8 9,10/5&ka

ye pi! galena bha%it " v" sukim"%davyacchandask "r "bhy"* |tata# stoka* vak - y"mi ch"todari -a+ prak "rai# || (6.1)

“O Slim-waisted, I will describe the gist of what was taught by Pi! gala and the prosodistsV " suki and M "%davya by means of six procedures.”

prast "r " ye sarve na -+ oddi -+ a* tathaiva laghukriy"m |

sa*khy"madhv"na* caiva ch"todari tatsphuta* bha%"ma# || (6.2)

“O Slim-waisted, we reveal all those, prast "ra, na -+ a*, uddi -+ a*, laghukriy", sa*khy" andadhv".”

s)cirmerupat "k " samudravipar ( tajaladhip"t "l "# |tath" $"lmalipras+"ra# sahito vipar ( ta $"lmalin" || (6.6)

“Thus, S )ci, Meru, Pat "k " , Samudra, Vipar ( ta-jaladhi, P "t "la, '"lmali and Vipar ( ta- $"lmali pras+"ra.”

Virah"!ka describes 8 different ways of enumerating meter variations. The first two are the twoversions of Lagakriy". S )ci is Pi! gala’s Lagakriy" while meru construction is exactly what isnow called the Pascal’s triangle. Pat "k " and Samudra are the two methods of listing all theforms of a meter given by Bharata. Virah"!ka also tells us how to modify these in order toenumerate the forms of M "trameters. Vipar ( ta-jaladhi is the procedure for listing the forms inreverse order. P "t "la lists the total number of permutations of a meter, the combined number ofsyllables in all the permuations, the combined number of m"tr "s, the combined number of Lsand the combined number of G’s. '"lmali tabulates in each line the number of Ls, the number of


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"C

syllables and the number of G’s in each of the variaions of a M!tra-v' tta. Vipar ( ta- $"lmali produces the same table in reverse order.

ma%iravam"l "k "ro dvigu%advigu%airvardhita# krama $a# | sth" payitavya# prast "ro nidhan"rdhama%( rav"rdha $ca || (6.13)

( Pat "k ") “ Prast "ra is established by (first) making a garland of G, L (that is, create a column ofalternating G, L.), (then) repeatedly doubling step-by-step, (that is, the second column of

alternating pairs GG, LL the third column of alternating quadurplets of GGGG, LLLL and soon), half of the syllables put down are G, and half are L.”

dvit ( y"rdhe $u kv" pi d ( yate spar $o’pyantima $chate |teneya* prast "re v, tt "n"* kriyate ga%ana || (6.14)

“O slim one, in the second half, place L everywhere as the last syllable. In this way, construct the prast "ra of v, tta-meters.”

ratn"ni yathecchay" sth" payitv" mugdhe sth" paya prast "ra* |t "vacca pi%0 aya sphuta* spar $"# sarve sthit " y"vat || (6.15)

(Samudra) “O bewildered! After placing as many G as required, establish the prast "ra byassembling (forms) until (the form with) L placed is seen everywhere.”

prathamacamarasy"dha# spar $a# purato yath"krame%aiva |marge ye pari $i -+"# ka+ akaist "n p)raya || (6.16)

“Successively (line-by-line), (put down) L below the first G; after it, the same as the line above;fill the remaining portion of the line with Gs.”

e -a eva prast "ro m"tr "v, tt "n"* s"dhita# ki*tu |m"tr " yatra na p)ryate prathama* spar $a* tatra dehi || (6.20)

“ prast "ra of m"tr "meters also is achieved similarly, except that whenever a form does not havethe full complement of m"tr "s, supply (the necessary number of extra) Ls at the beginning of theform.”

By this method, we get the following forms for the 4-m!tr ! foot:

GG

LLG

LGL

GLL

LLLL

Here, the italicized L is the L supplied to make up for the m!tr ! deficit.


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)*+*: ;0/5


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""

guroradhast "llghum"dita# k $ipetpara* likhed )rdhvasama* punastath" | p"$c"tyakha%da* guru%" prap)rayedy"vatpada* sarvalaghutvam" pyate || (8.2)

“First, write L below the first G, followed by a copy of what is above. Fill the initial portion

(before the newly written L) with Gs. Repeat this until the form with all L is obtained.”

vinyasya sarvagurvekadvicatu -k "nsam"rdhasamavi -am"*hr ( n | prast "rayetp, thakp, thagiti kram"t v, ttavidhiraya* prast "ra# || (8.3)

“After writing down one, two or 4 feet consisting of Gs in the case of sama, ardhasama orvi -ama meter, enumerate by repeating thus (the procedure described above) step by step. This isthe prast "ra for the entire meter.”

ekaiken"ntarit " prast "raprathamapa!ktiriha gurulaghun" |tattaddvigu%itagalata# kram"ddvit ( y"dipa!ktayo’ntarit "# syu# || (8.4)

(Alternate prast "ra) “The first column of the enumeration consists of alternating G and L. Theother columns beginning with the second are filled by alternating groups of Gs and Ls, thenumber of syllables in each group twice the number in the previous column.”

ardhasamaprast "re sa*d ,$ yante sam"rdhasamav, tt "ni |vi -amaprast "re’tra tu sam"rdhasamavi -amav, ttar ) p"% yakhila* || (8.5)

“The ardhasama forms enumerated in this way include the sama forms. The vi -ama formsinclude all the sama and ardhasama forms.”

j"t ( n"mapi catura# p"d "nvinyasya sa*bhavatsarvagur )n | prast "rayediti pr " gga%asa*bhavamapi laghuprayoga* jñ"tv" || (8.6)

“Even in the case of m!tr !meters, the clever after writing down feet with all G’s, carries out prast "ra knowing that (they have to) occasionally employ an initial L.”

)*+*D E@?510

p"de sarvagur "v"dy"llaghu* nyasya guroradha# | yathopari tath" $e -a* bh) ya# kury"damu* vidhi* || (6.1)

)ne dady"d gur )% yeva y"vatsarvalaghurbhavet | prast "ro’ya* sam"khy"ta $chandovicitivedibhi# || (6.2)

(forms of a single foot). “Below the first G of the foot consisting of all Gs, put down L.Repeatedly, make the rest same as what is above. This is the procedure. Supply Gs when missing

(syllables) (i.e. before the L.) (Continue) until (the form consisting of) all Ls is created. Theexperts in Chanda#$" stra call this prast "ra.”


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"$

)*+*F G@H0I04?10

atha prast "r "daya# -a+ pratyay"# || (8.1)

“Now the six algorithms beginning with prast "ra.”

pr "kkalp"dyago’dho la# paramuparisama* pr "k p)rvavidhiriti samayabhedak , dvarja* prast "ra# || (8.2)

“Below the first G of the previous form, (place) L. Beyond, the same as above. Before (the justwritten L), same procedure as before (i.e. write Gs). Avoid (forms) which differ from the rules

(For example, when treating ardhasama forms by this method, avoid sama forms which are alsocreated during the process.) (Thus is) prast "ra.”

gl "vadho’dho dvirdvirata# || (8.3)

“The pair G L repeatedly (copied) one below the other. Thereafter, double repeatedly.”

(This is the alternate version: The first column consists of alternating G L, the nextalternating GG LL and so on.)

?", J# '( A )

)*-*+ !, & gala

lardhe|(8.24) “In case of half, L.”

In case the given number can be halved, put L.

saike g | (8.25) “In case (combined) with one, G.”

In case it is necessary to add 1 in order to halve, put G.

Example: n = 6

6 is even, put L and halve it

3 is odd, add 1 before halving, put G2 is even, put L, halve it.

1 is odd, add 1 before halving, put G

When you reach one, the algorithm produces a series of Gs and is continued until the requisitenumber of syllables has been obtained. Thus, for meters of length 3, 4 and 5, you get LGL,

LGLG and LGLGG respectively in the sixth position.

)*-*- ./01020

v, ttasya parim"ñantu chitv"rdhena yath"kramam |nyasellaghu tath" saika* ak -ara* guru c" pyatha || (128)


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"%

“according to the rule, when dividing the meter’s measure (serial number) into half, put down L,

but with one (that is, if one is added to be able to halve the number), syllable G.”

)*-*) 304561070na -+ a* (6.10) “na -+ a*”

na -+ amidan(* vak - yate | h, tv" hordha* yasya kasyacit chandasa# sa*bhavati da $a $ata* sahasratama* v" dar $ayetyukte tenokt " y" sa*khy" t "vanti r ) p"%i vinyasya tadardhamapan( yalaghumeka* nyasya puna# punareva* k "rya# |

“Now, let na -+ a* be told. Dividing into half, (put down) L, when asked to show which of theforms occurs as the tenth or the hundredth or the thousandth (in the listing), Putting down thedigits of the given number, obtain its half, put down one L and repeat again and again.”

datvaika* same bh"# (6.11) “When (made) even after providing one, Gs.”

ardhe punarhniyama%e yadi vi -amat " sy"t tata evaika* dattv" samat "* k , tv" tato’rdhamapan( ya gurumeka* vinyasya puna# punareva* k "rya# | y"vaddid , ak -itasyachandaso’k -ar "%i parip)r %"ni bhavanti t "vadeva* k , tv" ida* tadv, ttamiti dar $ayet |

“If during repeated halving, there occurs oddness (odd number), then creating evenness by

giving (adding) one, after that, obtain half, put down one G and repeat again and again. Continueuntil the desired syllables of the meter have been competed, thus that form is to be exhibited.”

)*-*8 9,10/5&kaet "vatya!ke kataradv, ttamiti na -+ a* bhavati |tajj"n( hi v, tta* katame sth"na ityuddi -+ a* || (6.30)vi -am"!ke -u ca c"mara* same -u spar -a* sth" paya v, tt "n"* |ardhamardhamava -va -kate na -+"!ke sarvaka+ ak "ni || (6.31) yatra ca na dad "ti bh" gameka* datv" tatra pi!daya |bh" ge datte ca sphu+ a* m, g "k -i na -+ a* vij"n( hi || (6.32)

“ Na -+ a* is (the question): What is the form, given such a number (that is, its serial number)?(Conversely), uddi -+ a* is: if you know the form, what is its place (in the listing)?When the form’s number is odd, put down G, L when even. (The first compund word of the

second line is somewhat garbled, but the commenrary makes the meaning clear.) Halvingrepeatedly until number 1 is reached, (fill the the rest of the form with) all Gs. (In the above

procedure) whenever it is not possible to halve, (that is, the number is odd), add one and thenhalve. Know the true na -+ a*, O deer-eyed!”


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)*-*: ;0/5


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"?

na -+"!ke ga%airh, te $e -asa!khyo ga%o deyor "$i $e -e labdha* saika* || (8.5)

( Na -+ a* in the case of ga%achanda#) “Given the serial number of a missing form, repeatedly

divide by (the number of forms of its individual) ga%as (feet), and write down the form of the ga%a whose serial number is given by the remainder. (If the remainder is zero, it is considered to be equivalent to the divisor.) (If the remainder is not zero), add one (to the quotient).”

?"? KLLM '( # )

)*)*+ !, & gala

pratiloma-ga%am dvi# l "dyam (8.26): “opposite direction times two L first.”

Proceeding from right to left (and with initial value one), starting at first L (successively) double.

(here, ganam should be gunam.)

tata# gi eka* jahy"t (8.27): “(However,) there, if G (is encountered), subtract one (afterdoubling).”

Example: LGLG

Initially, n=1.start at the last L: LGLG, get n = 2

The next letter is G: LGLG, Get n = 2x2-1=3.The next letter is L: LGLG, Get n = 2x3=6.

Done

)*)*- ./01020

anty"d dvigu%it "drup"d dvidvirekam gurorbhavet |dvigu%"ñca lagho# k , tv" sa*khy"* pi%0 ena yojayet || (121)

“From the end, (starting from) one multiplied by two, repeatedly doubled, remove one from Gs(doubling).”

“From multiplication by two, obtaining (numbers associated with) L, calculate the (serial)

number by aggregating.”

The verse clearly describes Uddi -+ a*. Abhinavagupta treats this as single algorithm, but the twolines clearly describe two different versions. The first line is a somewhat simplified version of

Pi! gala’s algorithm except that bhavet does not fit. Alsdorf’s version has the word haret insteadwhich make sense. The second line may be interpreted as an abbreviated alternative algorithm,

described more fully by Mah"vira (see below.) In fact, instead of translating the verse as given, Abhinavagupta merely substitutes the full algorithm as given by Mah"vira, Jayadeva and Jayak ( rti and it runs as follows: (starting at the beginning of the verse), multiply one by two and


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"@

then repeatedly multiply by two, delete the numbers associated with Gs in the verse, addnumbers associated with Ls, finally, extend the sum (by one). (In Bharata’s version, the last step

is missing.) For example, consider the combination LGLG. We get the sequence 1,2,4,8. Throwout 2 and 8, add 1 and 4: we get 5. Increase 5 by 1 to get 6 which is the index of LGLG.

eva* vinyasya v, tt "n"* na -taoddi -tavibh" gatah | gurulaghvak -ar "n( ha sarvachandassu dar $ayet || (129)

“By putting down here GL syllables of forms in the case of all meters by appropriate means, Na -ta* or Uddi -ta*, exhibit (them).”

This is perhaps to suggest an alternate way to list all the forms of a meter as later described by

Mah"v( ra (see below.) Simply apply na -ta* to all the indices serially.

)*)*) 304561070

uddi -+ a* (6.12) “uddi -+ a*”

uddi -+ amid "n(* vak - yate | “Now, let uddi -+ a* be told.”

dvigu%a* dvigu%a* vardhayetpratilomata# | (6.13)kasyacicchandaso yatkiñcidv, tta* vinyasyeda* v, tta* katamadityukte tasya v, ttasy"-nty"k -ar "d "rabhya dvi catv"rya -tau -oda $a iti pratilomato dvigu%a* dvigu%a* vardhayet |

“Increase by repeatedly multiplying by two in reverse manner (from the end to the beginning.)

Put down some form of a meter. When asked after writing down some form of a meter, whichform (in the enumeration) it is, beginning at the last syllable, increase (one), (to) two, four, eight,

etc. by repeatedly multiplying by two proceeding in reverse order.”

ekah"nirbhe | (6.14) “Subtract one in case of G.”

eva* pratilomato vardhayato gurau vardhite ekena h"ni# k "ry" | eva* k , tv" yath"labdhasa*khy"va $"dvaktavyamiti |

“Going in reverse order in this way, in case of the long syllable, after increasing (by doubling),

subtract one. Having done this, announce the number thus obtained (as the serial number of theform).”

ayamekast ) pade $a# (6.15) “Here (is) one more instruction (lesson).”

tatra sarve -"* chandas"* v, tte -u vidhivat prastarite -u tatr "dyamardha* gurvanta* bhavati |aparamardha* laghvanta* | tatra laghvante uddi -+ e v, tte uttar "rdhe v, ttamiti jñ"tv" tasyottar "-rdhav, ttasy"nty"k -ar "t prabh, ti dvigu%a* vardhite p)rvavadekah"n(* k , tv" "chando’k -araparisam" pte# p)rv"rdhav, tt "ni ca vaktavy"n( ti | aya* t , t ( yasya krama# |


28/44

"A

“( Jan"$raya now attempts to explain the algorithm.) In all forms of a meter, when listedaccording to the algorithm, there, the first half has G at the end. The other half (has) L at the end.

Therefore, when an L-ending form is indicated, realizing it as a form in the second half, beginning from the last syllable of the form from the second half, multiplying by two, (and) in

case of G, subtracting one after doubling as before, until the end of (all) the syllables of the

meter. The forms of the first half to be spoken of similarly.”

)*)*8 9,10/5&ka

anta* spar $a* g , h( tv" dvigu%"ddvigu%e -u sutanu var !e -u |ekaika* camare -u muñcoddi -+ e ch"te || (6.34)

“O slender, O slender-waisted, when performing uddi -+ a*, starting from the last L, repeatedlydouble for each syllable, subtracting one if the syllable is a G.”

(Verses 6.36-6.40 deal with na -+ a* and uddi -+ a* of ga%achanda#. See Hemacandra below.)

)*)*: ;0/5


29/44

"B

“Above each G and L in a given form, let there be numbers (obtained by) repeatedly doublingthe intial 1. The sum of numbers in place of Ls with one added is the (location of) the form in the

enumeration.”

)*)*D E@?510uddi -+ a* dvigu%"n"dy"duparya!k "nsam"likhet |laghusth" ye ca tatr "!k " stai# saikairmi $ritairbhavet || (6.4)

“Starting from the beginning (of the form), write successively double numbers above (syllablesof) the entire form. The sum of those numbers which are above Ls togetherwith one

is the uddi -+ a* (serial number).”

)*)*F G@H0I04?10

uddi -+ e’nty"ll "ddvirgeka* tyajet || (8.6)

“To find the serial number of a given form, starting with the last L in the form, (going in reverse

order), successively double (the initial one), subtracting one when G (is encounterd).”

"dyamantyena hata* vyadhastana* || (8.7)

“(This verse deals with ga%achanda#.) Starting from the end, successively multiply (the numberof forms of each ga%a), subtracting (after each multiplication) the number of forms of the ga%a following (the given form of the ga"a in the list of all forms of the ga%a).”

?"A N#O#E"M%(

)*8*+ !, & gala

As mentioned in the Introduction, Hal " yudha interprets the repetition of the s)tra “ pare p)r !a*iti” as an instruction for the construction of meru-prast "ra. This seems to me very far-fetched. (See the section on Sa!khy" below.) The following s)tra quoted by Weber occurs in theYajur recension of the Chanda#$" stra, but not in its 1k recension nor in Hal " yudha’s version.

ekottarakrama $a# | p)rvap, kt " lasa*khy" ||

“Increasing by one, step-by-step, augmented by the next, L-count.”

This occurs just before the s)tra (8.24) in Hal " yudha. As discussed in Section 2.4, Weber cannotmake any sense out of this. The s)tra is most likely a cryptic lagakriy". It has the elements ofthe algorithm given more fully Bharata below. For a fuller comparison, see Section 2.4.


30/44

$C

)*8*- ./01020

ek "dhik "* tath" sa*khy"* chandaso vinive $ ya tu | y"vat p)rñantu p)rveña p)rayeduttara* gaña* || (124)eva* k , tv" tu sarve -"* pare -"* p)rvap)raña* |kram"nnaidhanam ekaika* pratiloma* visarjayet || (126) sarve -"* chandas"meva* laghvak -aravini $cayam | j"n( ta samav, att "n"* sa*khy"* sa*k -epatastath" || (127)

“Put down (a sequence, repeatedly) increased by one upto to the number (of syllables) of themeter.

Also, add the next number to the previous sum until finished.Also after thus doing (the process of) addition of the next, (that is, formation of partial sums) of

all the further (sequences),Remove one by one, in reverse order, the terminal (number) successively.

Of all meters with (pre)determined (number of) short syllablesThus know concisely the number of sama forms”

(The first word in the texts of Regnaud and Nagar is ek "dik "* (numbers beginning with one)instead of ek "dhik "* (numbers increasing by one) which is what Alsdorf has.

)*8*) 304561070

laghupar ( k -" (6.16) “Investigation of short syllables”.

laghupar ( k -ed "n(* vak - yate | “Now the investigation of short syllables will be spoken of”.

ekaikav, ddham"cchandasam (6.17) “Increasing repeatedly by one until the end of the meter.”

yasya kasyacicchandasa ekalaghuv, tt "n"manye -"* ca pram"%a* jijñ" sam"neekaikav, ddh"nya-k -arasth"(na)ni did , k -ita* chando’k -arapram"ni kram"t nyasy"ni | eva* vinyasya p)rva* p)rva* parayutamavin"$ y"ntyamap" sya p)rvamak -arasth"na* pare%"k -arasth"nena yukta* kartavyam | avin"$ y-"ntyamak -rasth"na* p)rvamev" p" sy"nyatra nyasyet | tata# puna# puna# p)rva* parayuta* k , tv"nyatra nyastasya parata# parato nyasitavyam | antyamekamava $i /+ a* bhavati | tacca te -"* parato nyasitavyam | eva* vinyasya prathamak -arasth"namavek - yatadupalabdhasa*khy"va $"dekalaghuv, tt "n( yant ( ti vadet | eva* t , t ( y"d ( navek - yatrilaghuv, tt "d ( ni br ) y"dityayamanya# krama# |

“When it is desired to know the number of forms with one L or others (forms with some other

number of Ls) belonging to some meter class, put down sequentially (numbers) increasingrepeatedly by one, in a number of locations equal to the number of syllables in the meter.

Repeatedly, add next number to the previous (sum), keeping (these numbers, except) discardingthe last, (that is), combine the (content of) the location of the next syllable with (that of) the

location of the previous syllable. Write elsewhere the undestroyed (saved) numbers, havingremoved the last number. Without destroying the location of the last syllable, discarding the


31/44

$#

previous, put it down elsewhere. Then, put down further and further (columns of numbers) byadding again and again next to the previous and writing them elsewhere. Finally, (only the

number) one is left. That too is to be written after those (earlier numbers). Written thus,announce that so many forms with one L according to the number obtained by observing the first

location. Similarly, observing the third (location), speak of forms with 3 Ls and so on.

(Now), here is another algorithm”.

The remaining text in this section is unclear. It might be an algorithm to list the forms with a

specified number of Ls. (For example, Var "hamihira has such an algorithm: B, hatsa*hit ", Adhy" ya 76, Verse 22.) Here is the untranslated text.

sabindv"dya* tato dvidvi# sabindu# (6.18) sabindu adyamak -arasth"na* kury"t | pañc dvigu%advigu%it "ni did , k -ita* chandok -arapram"%"ni k "ry"ni | eva* k , tv" ekalaghuv, tt "didid , k -" y"* saty"* tadodhor ) pa* nayet |

kram"d dvirnayet (6.19)

te -"* tath" nyast "n"mak -arasth"n"madhast "ddid , k -italaghuv, ttaprm"%"ni r ) p"%i vinyasyakram"nnayet | eka* dve tr (%( ti ga%ayann"dy"d "rabhy"nt "deva* n( tv" yath"labdhasa*khy"-va $"dek "dilaghuv, tt "d ( ni vaktavy"nyet "vant ( tyanena sve -u r ) pe -u vinyaste -vantya* yath" sthitasth"namekaikamiti dve ityuktamak -arasth"na* nayediti | t "ni punarv, tt "n( da* dve iti jijñ" s" y"* saty"* yuktir ) p"%i vinyasya sar ) pa* bindun" vadet | saha r ) pe%a vartata iti sar ) pa* bindun" s"rdha* sar ) pa* viga%ayya yath"labdhasa*khy"va $"dida* cedida* cetivaktavya* | te te laghavaste -u kasmin kasmin sth"ne sthit " iti cet –

sth"n"ni t "nyeva (6.20)te -"* lagh)n"* sth"n"ni t "nyeva bhavanti | etadukta* bhavati – ye -u ye -vak -arasth"ne -ur ) p"% yavasthit "ni t "nyeva sth"n"ni v, tte -u lagh)n"miti | iyamapar " laghupar ( k -" | ukt " ek "da $asya v, ttasya laghava# kiyanta ityukta p)rvavadak -arasth"n"ni nyasya t "vat y"vadbhi# sabindubhi# sth"nairek "da $asa*khy" parip)r %" bhavati tãvantastasyaik "da $asya v, ttasyalaghavo bhavanti | eva* vi $e -"%"mapi jñeyam| atra tatsth"n"ni t "nyeva bhavanti |

)*8*8 9,10/5&ka

pramukhente ca ekaika* tathaiva madhya ekamabhyadhika* | pratham"d "rabhya vardhante sarv"%k "# || (6.7)

ekaikena bhajyate uparisthita* tathaiva | parip"+ y" muñcaikaika* s)ciprast "re || (6.8)

tatpi%0 yat "* nipu%a* y"vad dvit ( yamapy" gata* sth"na* | prast "rap"taga%an" laghukriy" labhyate sa*khy" || (6.9)

(S )ci prast "ra) “Put down the numeral 1, in the beginning, the end and in between (as many asthe number of syllables in the meter) and one more. Increase all the numbers starting with thefirst (as follows.)”


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$"

“One-by-one, add the number above (to the partial sum). In the S )ci prast!ra, successively leaveout (the last number) one-by-one.”

“The accumulation is complete when the second place is reached (until the number to be left out

of addition is in the second place.) Laghukriy" number is obtained by carrying out the

algorithm.” (Velankar interprets the last line of 6.9 to mean that all the numbers of Laghukriy" are to be added up to obtain the total number of forms of a meter.)

iha ko -+ akayordvayorvardhate adha# sthita* krame%aiva | pramukh"nte ekaika* tata $ca dvau traya $catv"ra# || (6.10)

uparisthit "!kena vardhate’dha# sthita* krame%aiva |merau bhavati ga%an" s)cy" e -a anuharati || (6.11)

s" garavar %e’ !kau dvaveva gur ) madhyamasth"ne | samare punareka eva merau tathaiva s)cy"* || (6.12)

( Meru) “Two cells (rectangles) in a place, successively increase (the number

of cells) below them. In the first and last cell (enter) numeral 1 in (rows) 2, 3, 4 (etc).”

“Step-by-step, in (each) cell below, (place) the sum of the numbers in the (two) cells above. Thecalculation of the S%ci prast!ra is (re)created in the (table called) meru (named after the mythical

mountain). This (procedure) imitates (it.)”

“In the case of odd number of syllables, there are two large(st) numbers in the middle, moreover,in the case of even number of syllables, there is only one (such) in the meru, just as in S )ci prast "ra.”

)*8*: ;0/5


33/44

$$

6

1= 6,

6 • 5

1• 2=15,

6 • 5 • 4

1• 2 • 3= 20, etc

)*8*> 3070?@


34/44

$%

)*8*F G@H0I04?10

var %asam"nekak "n saik "nuparyupari k -ipet|muktv"ntya* sarvaik "digalakriy" || (8.8)

“(Write down as many) numeral 1’s as the number of syllables plus one. Repeatedly add the next

number above, leaving out the last. This is lagakriy" for all (Ls) beginning with one.”

"dyabhed "nadho’dho nyasya parairhatv" gre k -ipet || (8.9)

“(Now the extension to ga%achanda#.) Write down the numbers corresponding to the types offorms of the first ga%a, one below the other. Multiply them with those of the second and add theresulting columns. (continue the process.)” (See Alsdorf for a translation of related Hemachandra’s commentary on this.)

)*: '# ! EG%( P S n Q

)*:*+ !, & gala

dvirardhe (8.28) “two in case of half.”

If n can be halved, write “twice”.

r ) pe $)nyam (8.29) “In case one (must be subtracted in order to halve), ‘zero’.”

dvi# $)nye (8.30) “(going in reverse order), twice if zero’.”

t "vadardhe tadgu%itam (8.31) “In case where the number can be halved, multiply by itself (thatis, square the result.)”

Example: n = 6First construct the second column in the table below. Second, going back up, construct the third

column:n=6 twice (2!22)2 = 64

n=3 zero 2!22 = 8

n=2 twice 22 = 4

n=1 zero 2

Thus, total number S 6 of possible forms of length 6 is 64.

dvird(v?))na* tadant "n"m (8.32)

“twice two-less that (quantity) replaces (the sequence of counts) ending (with the current

count).”

Twice sn minus 2 equals sum of the series ending with S n.


35/44

$&

2S n – 2 = S 1 + S 2 + !!! +Sn

pare p)r !a* (8.34) “next full”

pare p)rn2a*iti (8.35) “next full, and so on.”

That is, subsequent Sn’s are full double of the previous, without subtraction of 2.

Sn+1 = 2S n.

S )tra (8.35) is a repetition of s)tra (8.34). Its interpretation by Hal " yudha as an instruction toconstruct the modern-day Pascal’s triangle to implement “lagakriy"” makes no sense. On theother hand, repeating a word or a phrase is a common usage in Sanskrit to indicate repetition of

an action just described, namely, repeated doublings to determine the successive S n‘s.

To calculate sa%khy" of ardhasama and vi -ama forms, Pi! gala gives the following algorithmearlier in his composition. It is tempting to conjecture that his divide-and-conquer algorithm(8.28-31) is inspired by the algorithm below.

samamardhasama* vi -ama* ca (5.2)

“ sama (equal), ardhasama (half-equal), vi -ama (unequal).”

sama* tavatk , tva# k , tamardhasama* (5.3)

“By multiplying (the number of) sama by itself (one obtains) ardhasama.”

S )tra 5.5 below clarifies that one has to subtract the number of sama from this.

vi -ama* ca (5.4) “and (similarly, the number of) vi -ama.”

ra $ y)na* (5.5) “Subtract the quantity”

That is, subtract the quantity from its square. Therefore, the number of ardhasama is the square

of sama minus the sama. Similarly, the number of vi -ama is the twice-squared sama minus thesquare of sama.

Pi! gala has a single formula dealing with M "tr "chanda#:

s" g yena na sam" l "* gla iti (4.53)

“The (the number of) G (is equal to that quantity) by which the number of syllables is unequal

from the number of m"tr "s.” The formula is: #G = #m!tr ! - #syllable.


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$?

)*:*- ./01020

"dya* sarvaguru jñeya* v, ttantu samasa* jñitam |ko $a* tu sarvalaghvantya* mi $rarup"ñi sarvata# || (122)

Abhinavagupta interprets this verse as a brief description of Lagakriy" and calculation ofSa!khy" (total number of forms of a meter) therefrom. Literal translation of the verse suggeststhat this is merely a simple characterization of prast "ras. The verse reads as follows, with“ko $a*” replaced by “ko $e” as in Alsdorf’s version: “In the tabulation, sama forms are known tohave all Gs at the beginning, all Ls at the end, mixed syllables everywhere (else).” The verseseems to a prelude to the next verse dealing with ardhasama and vi -ama forms.

v, tt "n"ntu sam"n"* sa*khy"* sa* yojya t "vat ( m |r "$ y)namardhavi -am"* sam" s"dabhinirdi $et || (123)

“(Replacing ardhavi(am!) by ardhasam!), the verse reads:) The number of sama forms, aftermultiplying by as much (squaring it), less the (original) quantity, precisely specifies the number

of ardhasama.”

sam"n"* vi -am"ñ"* ca sa3 guñayya tath" sphu+ am |r "$ y)namabhij"n( yadvi -am"ñ"* sam" sata# || (125)

“(vi -am"ñ"* in the first line should clearly read ardhasam"n"*. With that change, the versereads:) (The number) of vi -ama forms is known precisely by what becomes evident aftersubtracting the original quantity from the multiplication (by itself) of the sum of the number of

sama forms and the number of ardhasama forms.”

)*:*) 304561070 sa*khy" (6.21) “ sa*khy" (total number of forms)”

sa*khyed "n(* vak - yate | yasya kasyacicchandasa# samav, ttasa*khy"didid , ak -" y"* saty"* tasya p"d "k -ar "%i vinyasya tata# -

“Let us now speak about sa*khy". When it is desired to know sa*khy" of sama forms of ameter, after putting down the (number of) syllables,” -

bh"rdhah, te (6.22) “G when divided into half”

vinyast "n"* p"d "k -ar "%"mardha* h, tv" guruny" sa# puna# punarevak "rya# | ardhe punarhniyam"%e yadi vi -amat " sy"t –

“dividing the number of syllables into half, having put down G, repeat the same again and again.If, when trying to divide into half, oddness occurs, -”

ho samamap" syaikam (6.23) “L (when made) even (by) subtracting one.”


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$@

ekamap" sya p)rvanyastasy"dho laghu* nyasaiva* sarv"% yapanayet | e -a sa*khy" garbha# |

“Subtracting one, putting down L below what was put down before, (and thus) divide all

(numbers.) This is the essence (meaning) of sa*khy".”

bhe dvi# (6.24) “In case of G, twice.”

laghau laghau dvidvi kury"t | t "vat " bhe gu%ayet vardhyedityartha# | uktamev"rtha* nir ) payi - y"ma# |

“Whenever (you have) L, double. In the case of G, multiply that quantity by itself, thus increase

(the number.) This said, we will illustrate its meaning.”

The rest of the commentary on this s%tra shows that this procedure yields the number of sama forms of the g " yatri meter which consists of 4 feet of 6 syllables each as 64.

dy)na* tadant "n"m (6.25) “two-less replaces ending with that.”

taddvirityanuvartate tad g " yatr ( samav, ttasa*khy" pram"%a* dvigu%( k , ta* ca dv"bhy"* h( na* tadant "n"* g " yatryant "n"* -a%%"* samav, ttasa*khy" pram"%a* bhavati | tattu /advi*$atyadhi- ka $atameva* vi $e -"%"mapi jñeyam | samav, tt "dhigame br )ma# - samav, ttovargam)lo g "dhv-ardhasama# | samatulena gu%ito varga ityucyate | r " gi%a# samavargastasyam)le g "$cedardh- samav, tt "dhikara%a* bhavati | samavargastu catv"ri sahasr "%i -a%%avati $ca sa tum)lona $catv"ri sahasra%i dv"tri*$adadhik "ni |

“follow (the rule) ‘twice that’. (The s)tra obviously refers to Pi! gala’s original s)tra.)Thus, after doubling the sa*khy" of sama forms of the g " yatri meter and subtracting two from it,the (total) number of ( sama forms of the first) six meters ending with the g " yatri is obtained.Thus, 126 is known as sa*khy" in this particular case. Having obtained (the number) for sama forms, we now say: square of (number of) sama forms; by subtracting the original, (get) (number

of) ardhasama forms. Square is defined as multiplying by the equal (the same) quantity. Theformula for ardhasama is the square of sama (forms) of r " gin ( g " yatri) (less the original(quantity). (In the case of the g " yatri meter), square of sama is 4096, that less the original is4032.”

ubhayavargo vi -ama# (6.26) “twice square vi -ama”

m)lona iti vartate | ubhaye -"* sam"n"* v, tt "n"* varga# sa tu m)lona# vi -ama v, ttapram"%a* bhavati | ubhayavarga# kiy"niti cet pratiloma eka# -a+ sapt "tha sapta dve caikameva -advarga# sam"rdhasamyorg " yatr ( ya kathayedbudha# m)la* tu catv"ri sahasr "%i -a%%avati $ca tena h( naubhayavarga# |eka* -a+ sapta sapt "tha tr (% yeka* dve ca $)nyakam | pram"%a* vi -am"%"* tu g " yatry" lak -ayedbudha# ||evamanye -"* chandas"* jñeyam |


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$A

“(The rule) ‘Less square root’ (still) applies. Subtraction of the square root from the twice-

squred sama yields the number of vi -ama forms. If asked how much is twice square, in reverseorder: one six seven and two more sevens one and also six (16777216), the square of sama and

ardhasama of g " yatri, the experts say; the square root is 4096, twice-square less that (square

root): one six seven seven again three one two and zero (16773120); the expers recognize (this)as the the measure (number) of vi -ama forms of the g " yatri meter.(The number of forms) of other meters are found similarly.”

" sam)hairyath" svamalpabhed "nni - pann"n"* vait "l ( y"d ( n"* j"ti $lok "n"* l " ghave m"tr " yena pram"%ena y"vat "! gen"k -arebhyo’dhik " bhavat ( ti |

“In m"tr "meters, vait "l ( ya etc, constructed by means feet of 4 m"tr "s, namely, ga! g ", kurute,vibh"ti, s"tava, nacarati, and also feet consisting of 6, 7 and 8 m"tr "s, number of m"tr "s equalsthe amount by which twice the number of syllables exceeds the number of short syllables.”

)*:*8 9,10/5&ka

chando y"vatsa*khya* sth" payitv" sth" paya tasya p"d "!ka* |anenaiva gu%iten"rdhena bhavanti gurulaghava# || (6.41)

“After putting down the number (of forms of the foot) of a meter, put down the number ofsyllables in the foot. The product of the two numbers divided by 2 yields the total number of Gs

and Ls.”

k , tv" var %aga%ana* m"tr " bhavanti y" adhik "# |te gurava# $e -"# punarlaghava# sarv" su j"ti -u || (6.45)

“The number of Gs equals the excess of m"tr "s over the number of syllables. Moreover, thenumber of Ls the remainder among all the syllables.”

#G = #M!tr !s - #Syllables; #L = #Syllables - #G

antimavar %"ddvigu%a* var %e var %e [ca] dvigu%a* ku[ruta] | p"d "k -araparim"%a* sa*khy" y" e -a nirde $a# || (6.46)

“Starting from the last syllable, double (the initial one) for each syllable upto the number of

syllables in the meter. This shows the total count (of all the forms of a meter.)”

eva* ca var %av, tte m"tr "v, tt "n"manyath" bhavati |dvau dvau p)rvavikalpau y" m( layitv" j" yate sa*khy" | s" uttaram"tr "%"* sa*khy" y" e -a ni(r)de $a# || (6.49)

“Thus for the case of syallable-(based)-forms, but it is different for the m"tr "-(based)-forms. Thecount (of all possible forms of a m"tr "meter) is obtained by adding (the counts of) permutations


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$B

of the two previous (m"tr "meters). This is the way to the count (of total permuations) ofsucceeding m"tr "s (m"tr "meters).”

Thus, we get the formula S n= S n-1+ S n-2 which generates what is now known as the Fibonacci

sequence.

)*:*: ;0/5


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(and) multiplied by itself, is the count of vi -ama forms after the original (that is, the quantity thatwas squared) is subtracted.”

j"tya*hricatu -ke pratiga%a* k -ipettatra sa*bhavadga%asa*khy"* | gu%a(+ye)danyonya* tatsa*khy" sy"tsarvaj"tisa*khyeti mat " || (8.14)

“In the case of m"tr "meters with four feet, write down the number of possible forms for eachfoot. Multiply these numbers. The product is the approved sa*khy" of m"tr "meters.”

j"term"tr " pi%de sv"k -ararahite k , te sthit " gurava# syu# | gururahite’k -arasa*khy" laghurahite’rdh( k , te tu gurusa*khy" sy"t || (8.16)

“In the case of m"tr "meters, number of m"tr "s minus the number of syllables is (the number of)Gs. The number of m"tr "s minus the number of Gs (is) the number of syllables. The number ofm"tr "s minus the number of Ls, halved, gives the number of Gs.”

)*:*D E@?510

laghukriy"!kasa*dohe bhavetsa*khy" vimi $rite |uddi -+"!kasam"h"ra# saiko v" janayedim"* || (6.7)

“Sa*khy" is obtained by adding together the numbers obtatained by lagakriy". Alternatively, itmay be obtained by adding one to the sum of uddi -+ a* numbers.”

)*:*F G@H0I04?10

te pi%dit

"# sa

*khy

"# || (8.10)

“Sa*khy" is the sum of those (obtained by lagakriy").”

var %asamadvikahati# samasya || (8.11)

“(Alternatively) (The number) of sama forms (is) the product of as many 2’s as there are numberof syllables.”

te dvigu%" dvih( n"# sarve || (8.12)

“That number multiplied by 2 (and then) reduced by 2 (is) all.” (That is, the number of all sama

forms of meters with syllables from 1 upto the number of syllables of the present meter.)

samak , t ( r "$ y)n" ardhasamasya || (8.12)

“Square of the sama forms minus the original quantity is the number of ardhasama forms.”

tatk , tirvi -amasya || (8.14)


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%#

“Square of that (minus the original quantity) is the number of vi -ama forms.”

vikalpahatirm"tr "v, tt "n"* || (8.15)

“The number of forms of a m"tr "meter (is) the product of number of forms (of its individualfeet).”

a!k "ntyop"ntyayoga# pare pare m"tr "%"* (8.16)

“The sum of the last and the one before the last is the number of forms of the next m"tr "-foot.”

This is the formula S n=S n-1+S n-2 given by Virah"!ka earlier.

)*> #LG9#%RO# *

)*>*+ !, & gala Hal " yudha’s version of Pi! gala’s Chanda#$" stra does not include this 6th pratyaya, but Weberquotes the following s)tra from the Yajur recension of the Chanda#$" stra (occurring just beforethe s)tra corresponding to s)tra (8.24) in Hal " yudha):

a! gulap, thuhastada%0 akro $"# | yojana* ity adhv" ||

“a! gula ( finger), p, thu (palm), hasta (hand), da%0 a (staff), kro $" (shout), yojana, thus space.”

Weber also quote the following s%tra from the * k recension occurring between (8.32) and (8.33).

ekone’dhv"

“when one (is) subtracted, space”

Clearly, both refer to adhvayoga#. The Yajur version lists units of measurements and ends with“thus space” without telling us how to compute it. The 1k version gives the actual formula,“Space when one is subtracted”. That is, to obtain the amount of space required to write downall the forms of a particular meter, double the total number of forms and subtract one. The

formula allows the writer to leave space between successive forms.

)*>*- ./01020Bharata omits adhvayoga#.

)*>*) 304561070

adhv" |6.27) “Space.”


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%"

adhav" id "n(* vak - yate | “Now speak about space”

v, tta* dvirekonam | (6.28) “(The number of) forms twice less one.”

v, tt "ni dvigu%( k , t "ni ekena h( n"ni adhvapram"%a* bhavati | sarve -"* chandas"* tatra

g " yatry" adhvayoga* dar $ayi - y"ma# | asy"# sarv"%i v, tt "ni dvigu%( k , t "ni –t "ni tr % yatha pañcapi pañca catv"ri tattvata# |catv"ri tr %i c" pyeva* pramit "nya! gul "ni vai ||

“The measure of space is twice the (number of) forms less one. Among all meters, we will

illustrate the space calculation of the space for g!yatri. Multiply by two (the number of) all of itsforms (and subtract one): Indeed, they exactly measure two three’s five five four four three and

one (33554431.)”

a! gulama -tau yav" dv"da $"* gul "ni vitastihasto dvau hastau ki -kudhanu# dhanu# sahasre dvekro $a $catv"ra# kro $" yojanam |

“Two Eights (16) yavas (grains of barley) (equals) an a! gula (finger), twelve a! gulas measure ahasta (hand), two hastas (equal) a ki -kudhanu (forearm-bow), dhanu when two thousand(equals) kro $a (shout), four kro $as (equal) a yojana.”

(In the next paragraph in coverting 33,554,431 a! gulas converted into yojanas, a differentconversion table is implicitly assumed: 1 hasta = 24 a! gulas, 1 dhanu = 4 hastas, 1 yojana =8000 dhanus.)

tanya! gul "ni hastasya sa*khay" trayoda $alak -"%i mannuv(?)da $a $ata* caik " sapt "* gul "ni |dhanu# sa*khay" tr (%i lak -"%i catv"ryayut "ni nava sahasr "%i ca pañca $at "ni pañcavi*$ati $ca |vi*$ati $ca(?) hasta $caika# yojanasa*khay" traya $catv"ri*$adyojan"rdhayojana $ca dhanu -"* sahasr "%i pañcavi*$atyadhik "ni pañca $at "ni ca sapt "! gul "dhiko hasta $caika iti sabh"h" ca yen"-k -arebhyo’dhik " sa khalu gururbhavati |

(Dividing) by the number of a! gulas in a hasta, (33,554,431 a! gulas equal) 1,398,101 (hastas)and 7 a! gulas. (1,398,101 divided) by the number of hastas in a dhanu (equals) 349,525(dhanus) and 1 hasta. (Dividing) by the number of dhanus in a yojana, (the final result is) 43+ yojanas, 1525 dhanus, 1 hasta and 7 a! gulas. Thus, the length of that which is constructed withthe two syllables, G and L, is indeed obtained.”

A hasta of 24 a! gulas must measure at least a foot. Assuming that, the space requirement of1,398,101 hastas amounts to about 265 mi

A history of Pingala's combinatorics (Jayant Shah)

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Transcript of A history of Pingala's combinatorics (Jayant Shah)