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MATHEMATICS AND OTHER ŚĀSTRAS: PĀöINI AND PIðGALA M.D.SRINIVAS CENTRE FOR POLICY STUDIES [email protected]

Panini and Pingala

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MATHEMATICS AND OTHER ŚĀSTRAS:

PĀöINI AND PIðGALA

M.D.SRINIVAS CENTRE FOR POLICY STUDIES

[email protected]

DEVELOPMENT OF VYĀKARAöA OR ŚABDAŚĀSTRA

Pre-Pāõinian: Yāska’s Nirukta, Prātiśākhya Texts

Āpiśali, Indra, Kāśakçtsna, Śākañāyana, Vyādi, etc

Pāõini (c. 500 BCE): Aùñādhyāyī Sūtrapāñha, Dhātupāñha, Gaõapāñha

Kātyāyana: Vārttika, Pāli-vyākaraõa

Patañjali (c. 100 BCE): Mahābhāùya

Śarvavarman: Kātantra-vyākaraõa

Candragomin (c.450 CE): Cāndra-vyākaraõa

Devanandin (c.450): Jainendra-vyākaraõa

Bhartçhari (c. 450): Vākyapadīya, Mahābhāùya-dīpikā

Jayāditya, Vāmana (c. 600): Kāśikāvçtti

Jinendrabuddhi (c.900): Kāśikāvivaraõa-pañjikā or Nyāsa

Kaiyaña (c. 900): Mahābhāùya-pradīpa

Haradatta (c. 1000): Padamañjarī

Dharmakīrti (c.1000): Rūpāvatāra

Hemacandra (c. 1100): Siddhahaimacandra, etc

Vopadeva (c.1250): Mudgdhabodha

DEVELOPMENT OF VYĀKARAöA OR ŚABDAŚĀSTRA

Rāmacandra (c.1350): Prakriyākaumudī Nārāyaõa Bhaññātiri (c.1600): Prakriyāsarvasva Bhaññoji Dīkùita (c.1625): Siddhāntakaumudī, Prauóhamanoramā, Śabdakaustubha Kauõóabhañña (c.1650): Vaiyākaraõabhūùaõa Varadarāja (c.1650): Laghu-siddhāntakaumudī, Sāra-siddhāntakaumudī Nāgeśabhañña (c.1700): Mahābhāùya-pradipodyota, Bçhacchabdendu- śekhara,Vaiyākaraõa-siddhāntamañjūùā, Paribhāùenduśekhara Grammars of Other Languages Tamil: Tolkāppiyam (c.200 BCE), Vīrasolīyam (c.1200), Nannūl (c.1300) Kannada: Karnāñaka-bhāùābhūùaõa (c.1100), Śabdamaõidarpaõa (c.1200), Karnāñaka-śabdānuśāsana (c.1600) Telugu: Āndhra-śabdacintāmaõi (c.1100), Āndhrabhāùābhūùaõa (c.1250), Triliïga-śabdānuśāsana (c.1300) Pali: Kaccāyana-vyākaraõa, Saddalakkhaõa (c. 1150) Prākçta: Prākçta-prakāśa, Prākçta-śabdānuśāsana (c.1200) Persian: Pārasīprakāśa (c.1575)

ŚĀSTRAS: PRESENT SYSTEMATIC PROCEDURES

Most of the canonical texts on different disciplines (śāstras) in Indian tradition do not present a series of propositions; instead they present a series of rules, which serve to characterize and carry out systematic procedures to accomplish various ends.

These systematic procedures are generally referred to as vidhi, kriyā or prakriyā, sādhana, karma or parikarma, karaõa, upāya etc in different disciplines.

The rules are often formulated in the form of sūtras: According to Viùnudharmottarapurāõa (3.5.1): A sūtra has to be concise, unambiguous, pithy, comprehensive, shorn of irrelevancies and blemish-less.

अल्पाक्षरमसिन्दग्धं सारवद ्िव तोमुखम्। अस्तोभमनव सू ं सू िवदो िवदःु॥

Pāõini’s Aùñādhyāyī is acknowledged to be the paradigmatic example of a canonical text in Indian tradition. All other disciplines, especially mathematics, have been deeply influenced by its ingenious symbolic and technical devices, recursive and generative formalism and the system of conventions governing rule application and rule interaction.

PĀöINI AND EUCLID

“In Euclid’s geometry, propositions are derived from axioms with the help

of logical rules which are accepted as true. In Pāõini’s grammar, linguistic

forms are derived from grammatical elements with the help of rules which

were framed ad hoc (i.e. sūtras)....

Historically speaking, Pāõini’s method has occupied a place comparable to

that held by Euclid’s method in western thought. Scientific developments

have therefore taken different directions in India and in the West.... In

India, Pāõini’s perfection and ingenuity have rarely been matched outside

the realm of linguistics. Just as Plato reserved admission to his Academy

for geometricians, Indian scholars and philosophers are expected to have

first undergone a training in scientific linguistics....”

[J.F.Staal, Euclid and Pāõini, Philosophy East and West, 15, 1965, 99-116]

Note: The word “derived” means “demonstrated” in the case of Euclidean

Geometry; it means “generated” in the case of Pāõini’s Grammar (upapatti

and niùpatti)

ŚABDĀNUŚĀSANA: PĀöINI’S AúòĀDHYĀYĪ

अथ शब्दानशुासनम ्।

अनुशासनं कृित त्ययिवभागने ुत्पादनम ् त ाकरणेन साक्षाित् यत इित साक्षात् योजनम् । [अ म्भ ीय दीपोद् ोत ाख्या]

अथैतिस्मञ्शब्दोपदशेे सित क शब्दाना ं ितप ौ ितपदपाठः कतर् ः गौर ः पुरुषो हस्ती शकुिनमृर्गो ा ण इत्येवमादयः शब्दाः पिठत ाः। नेत्याह। अनभ्युपाय एव शब्दाना ं ितप ौ ितपदपाठः। एवं िह ूयते बृहस्पितिरन् ाय िद ं वषर्सह ं

ितपदो ानां शब्दानां शब्दपारायणं ोवाच नान्तं जगाम। [पात लमहाभाष्यम् पस्पशािह्नकम्]

Now, the instruction of utterances

Instruction, namely generation (of utterances) by using prakçti, pratyaya and other components, this is done by grammar, and that it is its direct purpose.

Valid utterances cannot be taught by pratipada-pātha (stating each of them individually). Bçhaspati tried to teach Indra valid utterances by pratipada-pāñha for thousand divine years, but reached nowhere near the end.

ŚABDĀNUŚĀSANA: PĀöINI’S AúòĀDHYĀYĪ

कथं तह मे शब्दाः ितप ाः। िकि त्सामान्यिवशेषवल्लक्षण ं वत्यर्म।् येनाल्पेन य ेन महतो महतः शब्दौघान् ितप ेरन् ।

क पुनस्तत् । उत्सगार्पवादौ । कि दतु्सगर्ः कतर् ःकि दपवादः। कथंजातीयकः पुनरुत्सगर्ः कतर् ः कथंजातीयकोऽपवादः। सामान्येनोत्सगर्ः कतर् ः। त था। कमर्ण्यण् (३। २ ।१)। तस्य िवशेषेणापवादः। त था। आतोऽनुपसग कः (३। २ ।३)।

[पात लमहाभाष्यम् पस्पशािह्नकम्]

How are these utterances to be known?

Some characterisation with what is general and particular is to be provided, by which, with little effort, great amount of utterances are known.

What is that characterisation? Utsarga (general) and Apavāda (special/exceptional) rules...

INDOLOGISTS ON PĀöINI’S AúòĀDHYĀYĪ

“Of particular interest is the stress laid on the ‘small number of primitive

elements’, themselves not used (i.e., themselves abstract) from which the

Sanskrit grammarians are said to derive ‘the infinite variety of actual

forms in use.’ ” [J.F.Staal on Francois Pons’ letter of 1740 (published 1743) in, A Reader on the Sanskrit Grammarians, MIT Press, 1972, p.30]

“The Descriptive Grammar of Sanskrit, which Pāõini brought to its perfection, is one of the greatest monuments of human intelligence and an indispensible model for the description of languages. ”

[L. Bloomfield, Review of Liebich, Konkordanz das Pāõini-Candra, Language, 5, 267-276, 1929]

INDOLOGISTS ON PĀöINI’S AúòĀDHYĀYĪ

“The idea that a language is based on a system of rules determining the interpretation of its infinitely many sentences is by no means novel. Well over a century ago, it was expressed with reasonable clarity by Wilhelm von Humboldt in his famous but rarely studied introduction to general linguistics (Humboldt 1836). His view that a language ‘makes infinite use of finite means’ and that a grammar must describe the process that makes this possible.. Pāõini’s grammar can be interpreted as a fragment of such a ‘generative grammar’ in essentially the contemporary sense of this term. ”

[N. Chomsky, Aspects of the Theory of Syntax, MIT Press, 1964, p.v]

“Modern linguistics acknowledges it as the most complete generative grammar of any language yet written and continues to adopt technical ideas from it ”.

[P.Kiparsky, Pāõinian Linguistics, in Encyclopaedia of Language and Linguistics, VI, 1994]

INDOLOGISTS ON PĀöINI’S AúòĀDHYĀYĪ

The algebraic formulation of Pāõini’s rules was not appreciated by the first Western students; they regarded the work as abstruse or artificial. ... The Western critique was muted and eventually turned into praise when modern schools of linguistics developed sophisticated notation systems of their own. Grammars that derive words and sentences from basic elements by a string of rules are obviously in greater need of symbolic code than paradigmatic or direct method practical grammars....

It is a sad observation that we did not learn more from Pāõini than we did, that we recognised the value and the spirit of his "artificial" and "abstruse" formulations only when we had independently constructed comparable systems. The Indian New Logic (navya-nyāya) had the same fate: only after Western mathematicians had developed a formal logic of their own and after this knowledge had reached a few Indologists, did the attitude towards the navya-nyāya school change from ridicule to respect.

H. Scharfe, Grammatical Literature, Wiesbaden 1977, p.112, 115.

INDOLOGISTS ON PĀöINI’S AúòĀDHYĀYĪ 

“Pāõini has composed a list of formulae called sūtra...serving to form

words and sentences from a given material of minimal elements...It

comprises both lists of primary elements, and a program for the

combination of these elements. These elements are the phonemes, the

roots, group of words sharing a grammatical feature, morphemes

(suffixes) having a meaning ...

The program is made up of operating rules as well as conventions

necessary for the application of the rules. It is composed in a true meta-

language very apt to its purpose, achieving the maximum brevity, which

makes it easy to memorize, and is the first and foremost example of the

formalization of the technical exposition in the universal history of

sciences. Because of its practical objective and form, it cannot be

compared with a systematic grammar of a European type. By contrast, its

resemblance to a modern computer program is striking. ”

[P S Filliozat: The Sanskrit Language: An Overview, Indica Books, Varanasi 2000 (French Edition 1992), p. 24]

INDOLOGISTS ON PĀöINI’S AúòĀDHYĀYĪ

“Pāõini's grammar is universally admired for its insightful analysis of

Sanskrit...Generative linguists for their part have marvelled especially at

its ingenious technical devices, and at intricate system of conventions

governing rule application and rule interaction that it presupposes, which

seem to uncannily anticipate ideas of modern linguistic theory (if only

because many of them were originally borrowed from Pāõini in the first

place.)...

The grammar has four distinct components:

1. Aùñādhyāyī: a system of about 4,000 grammatical rules

2. Śivasūtras: the inventory of phonological segments...

3. Dhātupāñha: a list of about 2,000 verbal roots...

4. Gaõapāñha: a list of 261 lists of lexical items ...

The grammar is a device that starts from meaning information... and

incrementally builds up a completely interpreted sentence.”

[P.Kiparsky, On the Architecture of Pāõini's Grammar, 2002]

ŚIVA-SŪTRAS AND PRATYĀHĀRAS

१ अइउण्। २ ऋलृक्। ३ एओङ्। ४ ऐऔच्। ५ हयवरट्। ६ लण्। ७ ञमङणनम्। ८ झभञ्। ९ घढधष्। १० जबगडदश्। ११खफचठथचटतव्। १२ कपय्। १३ शषसर्। १४ हल्॥

Each sūtra has a set of varõas followed by a marker (ö, K, ð, C, ò etc)

called the called the it varõa [एषाम ्अन्त्या इतः]

Pratyāhāras are formed by any of the varõas and an it which follows it. The pratyāhāra then stands for the class of varõas enclosed by them except for the intervening it-varõas.

aK stands for {a, i, u, ç, ë }. iK stands for { i, u, ç, ë }

aC stands for all the vowels. haL stands for all the consonants

In this way about 300 pratyāhāras are possible; Pāõini uses 42 of them.

Recent studies show that the Śiva-sūtras give an optimal encoding for these 42 partially ordered subsets of Sanskrit sounds.

METHOD OF AúòĀDHYĀYĪ

Pāõini’s Sūtras are mainly of the following types:

• Vidhi-sūtra: Operational rules

• Saüjñā-sūtra: Rules which introduce class names and establish

conventions regarding the use of terms

• Adhikāra-sūtra: Headings

• Paribhāùā-sūtra: Metarules, which serve to interpret and regulate other

rules. They regulate the operations specified in the vidhi-sūtras:

• úaùñhīsthāneyogā (1.1.49): Genitive designates ‘in place of’

• Tasminnitinirdiùñe pūrvasya (1.1.66): Locative defines the right

context

• Yathāsaükhyamanudeśaþ samānām (1.3.10): For groups with the

same number of elements, the corresponding elements are to be

related in order.

• Pūrvatrāsiddham (8.2.1): (From now on every rule is regarded as)

not having taken effect with reference to preceding ones.

CONTEXT SENSITIVE RULES OF AúòĀDHYĀYĪ

Example: Ikoyaõaci (6.1.77)

CONTEXT SENSITIVE RULES OF AúòĀDHYĀYĪ

ikoyaõaci (6.1.77)

iK stands for {i, u, ç, ë},

yaö stands for {y, v, r, l}

aC stands for all the vowels.

From 6.1.72, saühitāyām is carried forward. Thus the sūtra provides that:

i, u, ç, ë →y, v, r, l before a vowel, in close contact

This gives

i + a → y + a, u+ a → v + a, and so on.

Akaþ savarõe dīrghaþ (6.1.101) is an apavāda- sūtra to the above, and

gives:

i + i = ī, u+ u = ū, and so on.

PĀöINI AND ZERO

Panini introduces the notion of zero-replacement (zero-phoneme, zero-

morpheme etc)

Adarśanaü lopaþ (1.1.60) Non-appearance is zero.

There are about fifty sūtras where lopa appears explicitly and more than

hundred if we take into account anuvçtti.

There are several other kinds of zeroes in Panini.

For instance, there are the it varõas in pratyāhāras. [Tasya lopah (1.3.39)]

There are also luk, ślu and lup which correspond to non-appearance of a

pratyaya or suffix.

 

 

 

VĀKYAPADĪYA ON ŚĀSTRA AS UPĀYA

िभ ं दशर्नमाि त्य वहारोऽनुगम्यते। त यन्मुख्यमेकेषा ंत ान्येषा ंिवपयर्य:॥ (वाक्यपदीयम् १.७४) Worldly activities are accomplished on the basis of different theories and philosophies. What is important in one theory may not be so in another.

उपादायािप ये हयेा तानुपायान ् चक्षते। उपायाना िनयमो नावश्यमवित त े॥ अथ कथि द ्पुरुष: कथि त् ितप ते। (वाक्यपदीयम् २.३८-९)

Upāyas (procedures taught in śāstras) are to be discarded, even though they are to be used for accomplishing an objective. There is no necessary limitation on such means. One accomplishes objectives by one means or the other.

As noted by the commentator Puõyarāja:

कि दाचायर्ः पािणिनिवरिचतेन लक्षणशा ेण शब्दानिधगच्छित कि दन्येनेित न िनयम:।

DEVELOPMENT OF CHADAÿ-ŚĀSTRA

In his Chandaþ-śāstra (c.300 BCE), Piïgala introduces some combinatorial tools called pratyayas which can be employed to study the various possible meters in Sanskrit prosody. Following are some of the important texts which include a discussion of various pratyayas:

Piïgala (c.300 BCE): Chandaþ-śāstra Bharata (c.100 BCE): Nāñyaśāstra Brahmagupta (c.628): Brāhmasphuñasiddhānta Virahāïka (c.650): Vçttajātisamuccaya Mahāvīra (c.850): Gaõitasārasaïgraha Halāyudha (c.950): Mçtasañjīvanī Comm. on Piïgala’s Chandaþ-śāstra Kedārabhañña (c.1000): Vçttaratnākara Yādavaprakāśa (c.1000): Comm. on Piïgala’s Chandahþ-śāstra Hemacandra (c.1200): Chandonuśāsana Prākçta-paiïgala (c.1300) Nārāyaõapaõóita (c.1350): Gaõitakaumudī Damodara (C.1500): Vaõībhūùaõa Nārāyaõabhañña (c.1550): Nārāyaõī Comm. on Vçttaratnākara

VARöA-VèTTA

A syllable (akùara) is a vowel or a vowel with one more consonants preceding it.

A syllable is laghu (light) if it has a short vowel. Even a short syllable will be a guru if what follows is a conjunct consonant, an anusvāra or a visarga.

Otherwise the syllable is guru (heavy).

The last syllable of a foot of a metre is taken to be a guru optionally.

या सृि ः ुरा ा वहित िविधहुत ंया हिवयार् च हो ी ये ेकालं िवध ः ुितिवषयगुणा या िस्थता ाप्य िव म्। यामाहुः सवर्बीज कृितिरित यया ािणनः ाणवन्तः

त्यक्षािभः स स्तनुिभरवत ुवस्तािभर ािभरीशः॥

GGG GLG GLL LLL LGG LGG LGG

THE EIGHT GAöAS

आिदमध्यावसानेषु यरता यािन्त लाघवम्। भजसा गौरवं यािन्त मनौ त ुगुरुलाघवम्॥

Ya: LGG Ra: GLG Ta: GGL

Bha: GLL Ja: LGL Sa: LLG

Ma: GGG Na: LLL

The pattern of a metre is usually characterised in term of these gaõas.

For instance the verse of Kālidāsa cited earlier is in Sragdharā metre:

भ्नैयार्नां यणे ि मुिनयितयुता ग्धरा कीिततेयम्। Thus Sragdharā is characterised by the pattern: MaRaBhaNaYaYaYa, with a break (yati) after seven syllables each.

GGGGLGG LLLLLLG GLGGLGG

A MNEMONIC FOR THE GAöAS

There is the mnemonic attributed to Pāõini

YaMāTāRāJaBhāNaSaLaGam

L G G G L G L L L G

If we replace G by 0 and L by 1, we have a binary sequence of length 10

1 0 0 0 1 0 1 1 1 0

The above linear binary sequence generates all the 8 binary sequences of length 3. We can remove the last pair 1, 0 and view the rest as a cyclic binary sequence of length eight. In modern mathematics such sequences are referred to as de Bruijn cycles.

[।, ऽ stand for L, G or 1,0]

PRATYAYAS IN PIðGALA’S CHADAÿ-ŚĀSTRA

In Chapter Eight of Chandaþ-śāstra, Piïgala introduces the following six pratyayas:

Prastāra: A procedure by which all the possible metrical patterns with a given number of syllables are laid out sequentially as an array.

Saïkhyā: The process of finding total number of metrical patterns (or rows) in the prastāra.

Naùña: The process of finding for any row, with a given number, the corresponding metrical pattern in the prastāra.

Uddiùña: The process for finding, for any given metrical pattern, the corresponding row number in the prastāra.

Lagakriyā: The process of finding the number of metrical forms with a given number of Laghus (or Gurus).

Adhvayoga: The process of finding the space occupied by the prastāra.

PRASTĀRA

ि कौ ग्लौ। िम ौ च। पृथग्लािम ाः। वसवि काः। (छन्दःशा म् ८.२०-२३) Form a G, L pair. Write them one below the other. Insert on the right Gs and Ls. [Repeating the process] we have eight (vasavaþ) metric forms in the 3-syllable-prastāra

Single syllable prastāra

Two syllable prastāra  

 

 

1 G 2 L

1 G G 2 L G 3 G L 4 L L

PRASTĀRA

Three syllable prastāra  

 

Another method of generating the prastāra

पाद ेसवर्गुरावा ाल्लघुं न्यस्य गुरोरधः। यथोपिर तथा शेषं भूयः कुयार्दमु ंिविधम्। ऊन ेद ाद्गुरूनवे यावत्सवर्लघुभर्वते।् (वृ र ाकरम् ६.२-३)

Start with a row of Gs. Scan from the left to identify the first G. Place an L below that. The elements to the right are brought down as they are. All the places to the left are filled up by Gs. Go on till a row of only Ls is reached.

1 G G G 2 L G G 3 G L G 4 L L G 5 G G L 6 L G L 7 G L L 8 L L L

SAðKHYĀ

ि रध। रूपे शून्यम्। ि ःशून्ये। तावदध तद्गुिणतम्। (छन्दःशा म् ८.२८-३१)

The number of metres of n-syllables is Sn = 2n. Piïgala gives an optimal algorithm for finding 2n by means of multiplication and squaring operations much less than n in number.

Halve the number and mark "2" If the number cannot be halved deduct one and mark "0" [Proceed till you reach zero. Start with 1 and scan the sequence from the right] If "0", multiply by 2 If "2", square

Example: Six-syllable metres n = 6 • 6/2 = 3 and mark "2" • 3 cannot be halved (3-1) and mark "0" • 2/2 = 1 and mark "2" • 1-1 = 0 and mark "0"

Sequence 2, 0, 2, 0 yields 1x2, (1x2)2, (1x2)2x2, ((1x2)2x2) 2 = 26

Piïgala’s algorithm became the standard method for computing powers in Indian mathematics.

SAðKHYĀ

Next sūtra of Pingala gives the sum of all the saïkhyās Sr for r = 1, 2, ...n.

ि ूर्न ंतदन्तानाम।् (छन्दःशा म् ८.३२)

S1 + S2 + S3 + ... + Sn = 2Sn -1

Then comes the sūtra

परे पणूर्म।् (छन्दःशा म् ८.३३)

Sn+1 = 2Sn

Together, the two sutras imply

Sn = 2n

and

1 + 2 + 22 + ... + 2n = 2n+1 -1

The latter clearly is the formula for the sum of a geometric series.

SAðKHYĀ

The Saïkhya 2n discussed above is for the case of syllabic metres of n-syllables which are sama-vçttas − metres which have the same pattern in all the four pādas or quarters.

Ardhasama-vçttas are those metres which are not sama, but whose halves are the same. Viùama- vçttas are those whose pādas are all different.

In the fifth Chapter of Chandaþ-śāstra, Piïgala has dealt with the saïkhyā of Ardhasama and Viùama- vçttas.

समं तावत्कृत्वः कृतमधर्समम।् िवषम ंच। राश्यनूम्। ((छन्दःशा म् ५.३-५)

The number of Ardhasama-vçttas with n-syllables in each pāda is (2n)2 -2n

In the same way, the number of Viùama-vçttas with n-syllables in each pāda is [(2n)2 -2n]2 - [(2n)2 -2n]

NAúòA

लध। सैके ग्। (छन्दःशा म् ८.२४-२५)

• To find the metric pattern in a row of the prastāra, start with the row-number

• Halve it (if possible) and write an L • If it cannot be halved, add one and halve and write a G • Proceed till all the syllables of the metre are found.

Example: Find the 7th metrical form in a 4-syllable prastāra • (7+1)/2 = 4 Hence G • 4/2 = 2 Hence GL • 2/2 = 1 Hence GLL • (1+1)/2 = 1 Hence GLLG

If we set G=0 and L = 1, we can see that Piïgala’s naùña process leads to the desired metric form via the binary expansion

7 = 0 + 1x2 + 1 x 22 + 0 x 23

FOUR-SYLLABLE PRASTĀRA

1 G G G G 2 L G G G 3 G L G G 4 L L G G 5 G G L G 6 L G L G 7 G L L G 8 L L L G 9 G G G L 10 L G G L 11 G L G L 12 L L G L 13 G G L L 14 L G L L 15 G L L L 16 L L L L

If we set G=0 and L=1, then we see that each metric pattern is the mirror reflection of the binary representation of the associated row-number-1.

UDDIúòA

ितलोमगणं ि लार् म्। ततोग्येकं ज ात्। (छन्दःशा म् ८.२६-२७)

To find the row number of a given metric pattern, scan the pattern from the right.

Start with number 1 Double it when an L is encountered. Double and reduce by 1 when a G is encountered

Example: To find the row-number of the pattern GLLG in a 4-syllable prastāra:

• Start with 1. It is unchanged with first G • Then we find L. So we get 1x2 = 2 • Then we find L. So we get 2x2 = 4 • Finally we have G. We get 4x2-1 = 7

UDDIúòA

Another Method:

उि ंि गुणाना ादपुयर्ङ्कान ्समािलखेत्। लघुस्था ये तु त ाङ्कास्तैः सैकैिमि तैभर्वते्। (वृ र ाकर ६.५)

Place 1 on top of the left-most syllable of the given metrical pattern Double it at each step while moving right. Sum the numbers above L and add 1 to get the row-number

Example: To find the row-number of the pattern GLLG 1 2 22 23 G L L G

Row-Number = 0 x 1 + 1 x 2 + 1 x 22 + 0 x 23 + 1= 7

Both the naùña and uddiùña process of Piïgala are essentially based on the fact that every natural number has a unique binary representation: It can be uniquely represented as a sum of different saïkhyā Sn or the powers of 2.

LAGAKRIYĀ

परे पूणर्िमित। (छन्दःशा म् ८. ३४)

Piïgala’s sūtra on lagakriyā process is too brief. Halāyudha, the tenth century commentator explains it as giving the basic rule for the construction of the table of numbers which he refers to as the Meru-prastāra.

उपिर ादकंे चतुर को ं िलिखत्वा तस्यधस्तादभुयतोऽधर्िनष् ान्तं को क यं िलखेत्।

तस्याप्यधस्तात् यं तस्याप्यधस्ता तु यं यावदिभमत ं स्थानिमित मेरु स्तारः। तस्य

थमे को ेएकसंख्यां वस्थाप्य लक्षणिमद ं वतर्येत्। त परे को ेयद्वृ संख्याजात ं

तत ्पूवर्को योः पूण िनवेशयेत्। त ोभयोः को कयोरेकैकमङ्कं द ात ्मध्ये को ेतु

परको याङ्कमेकीकृत्य पूण िनवेशयेिदित पूणर्शब्दाथर्ः। चतुथ्या पङ् ाविप

पयर्न्तको योरेकैकमेव स्थापयेत्। मध्यमको योस्त ु परको याङ्कमेकीकृत्य पूण

ि संख्यारूपं स्थापयेत्।…

VARöA-MERU OF PIðGALA

Clearly the number of metrical forms with r Gurus (or Laghus) in the prastāra of metres of n-syllables is the binomial coefficient nCr

The above passage of Halāyudha shows that the basic rule for the construction of the above table, is the recurrence relation nCr

= n-1Cr-1 + n-1Cr

PASCAL TRIANGLE

The above Varõa-Meru is actually a rotated version of the so called Pascal Triangle (c. 1655) shown below:

REFERENCES

1. G.Cardona, Pāõini A Survey of Research, Mouton, The Hague 1976.

2. R.N.Sharma, The Aùñādhyāyī of Pāõini, 6 Volumes, Munshiram Manoharlal, Delhi 1990-2003.

3. G.Cardona, Pāõini His Work and its Traditions, 2ed, Motilal Banarsidas, New Delhi 1997

4. G.Huet and A.Kulkarni and P.Scharf Eds., Sanskrit Computational Linguistics, Springer, New York 2009

5. Chandaþśāstra of Piïgala with Comm. Mçtasañjīvanī of Halāyudha Bhañña, Ed. Kedarnath, 3ed. Bombay 1938.

6. Vçttaratnākara of Kedāra with Comms. Nārāyaõī and Setu, Ed. Madhusudana Sastri, Chaukhambha, Varanasi 1994.

7. B. van Nooten, Binary Numbers in Indian Antiquity, Jour. Ind. Phil. 21, 1993, pp.31-50.

8. R.Sridharan, Sanskrit Prosody, Piïgala Sūtras and Binary Arithmetic, in G.G.Emch et al Eds., Contributions to the History of Indian Mathematics, Hindustan Book Agency, Delhi 2005, pp. 33-62.