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Development of Combinatiorics from The Pratyayas in Sanskrit Prosody
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About the Author

Sri Venugopal D. Heroor is an engineer by profession, and a keen and enthusiastic scholar of Ancient Indian Mathematics. He has brought out Five books on History of Mathematics and Mathematician of Ancient and Medieval India.

He has translated the Sanskrit works: Sridhara’s Patiganita and Trisatika, Ganitadhyaya of Brahmagupta’s Brahmasputa Siddhanta, Bhaskaracary’s jyotpatti and Lilavati, Chitrabhanu’sEkavimsati Prasnottara, Narayana Pandita’s Ganita Kaumudu in to Kannada.

He has translated many articles and research papers and has also Contributed original articles.in the present book, Development of Combinatorics from the Pratyayas in Sanskrit Prosody’the topic of combinatorics i.e., Permutations, combinations, partitions, sequence of binomial and polynomial coefficients, binomial theorem, figures containing these sequences and relations between them dealt with earlier by Narayana Pandita in his Ganita Kaumudi is discussed in the light of Sanskrit Prosody in modern terminology.

 

Foreward

Unique is the way of knowledge. Wonderful is the journey of learning. In the annals of Humanb history every branches of knowledge has its own pulsating twist and turns that link with so many other branches of wisdom. It is for this reason, our masters have made dictum.

He who is initiated in only one branch of learning will be incapable of saying anything decisive in the same field.

Likewise it is also said that some of the branches of knowledge become very significant and indispensable to so many other systems of learning. In the same lines we can also boldly proclaim that language and Mathematics form the bedrock of all knowledge, past and present.

Here logic is equated with mathematics and this is understood by every serious student of it. Mathematics will never provide us with any ‘information’. It can only process it and bring out the details of the same in many ways that show wonderful internal relationships and equations. Likewise, language too, having the indispensable gramatic governance represents the needed information. But by itself, it cannot be that. However, thepath of mathematics I ssymbols while that of the language is sounds. But we do have the inseparable relationship between signs and sounds. In this way, the sequence of cause and effect, well found in language / grammar joins its hands with the same chain found in the counts and concepts of mathematics / logic.

Prosody is the science of metres that are employed in the art of versification. This essentially deals with the pattern of longer and shorter sounds /stressed and unstressed syllables. (similar to, but not essentially the same).In India we have a Great heritage for this science as it is one among the six limbs o f the Vedas. Since time immemorial, we had many verities of metres. In the vedic texts themselves we find several scores of metres which have their basic patterns in the famed seven groups. Coming to the classical period, the number of metres goes to several hundreds and the masters in this field have done a laudable job by giving beautiful classification and analysis. In this period perhaps, we see a great deal of work done on the mathematical relationship revealed in the patterns of longer and shorter sounds called as gurus and laghus – found in the matrix of any metre.

Here we should note that the three main divisions of Indian/Sanskrit Metres viz. the Varna Vrttas, the Matrajati-s and the Aksharajati-s / Visayajati-s have their own unique challenges to mathemaics and all these are will answered by the scholars in the respective fields. Thus in the finest art of poetry, Mathematics, the queen among sciences has made her way. But this has to do nothing with the aesthetics of prosody. Yet, we have to bear in the mind that the prosodic patterns alone create aesthetic effect and such a pattern has an inbuilt logic. However, these primary constructions are not engineered. They are essentially creations which sprang spontaneously. But again, we gave to say that such a spontaneous aesthetic creation has an inbuilt wisdom and awareness. Thus the whole issue leads us towards the inseparability of head and heart at the root. However, we are not going to those details here.

Our prosodists starting from Pingala, have given six folded scheme called satpratyaya prakriya to know some of the essential and mathematically tenable things in metrics. These pratyayas, called so because they reveal such details of prosody help us in presenting the possible metric patterns of guru laghus in one main set, search for us a lost pattern, locate a specific pattern, bring out the details of various metres, make us know the total number of verities in a set without going to the laborious process of prastara and hint at the possible space needed for a full-filedged presentation of the matrix. Among these, ekadvyadilagakriya is the most important and has many ways of accomplishing the same. Thus Mathematicians like Bhaskaracharya too have brought out their respective versions.

In this light, the present work of my esteemed learned friend Sri Venugopal D. Heroor has to be understood. Mr. Heroor has spared no pains in making his work, the ‘Development of Combinatorics from the pratyays in Sanskrit prosody’ both exhaustive and comprehensive. His literature survey is thorough and exposition is lucid. Since long, he has been in the field of Indian mathematics, which is indeed a hard nut to crack as it demands the knowledge of both logic and language! However, Mr. Heroor’s many earlier publications have carved a niche of his aim and this publication is going to become yet another feather in his cap. Therefore I wholeheartedly congratulate him and strongly recommend his work to the lare of Indic learning.

 

Preface

My study on the topic of ‘Combinatorics in Prosody’ started with the article ‘Use of Permutations and Combination in India’ by B. Datta and A. N. Singh, which I have translated into Kannada and published in the Kannada monthly UTTHANA, in June 2002. Later in December 2007, I gave a lecture on ‘The Concept of Binary place value system of numeration in ‘sat-pratyaya sutras of Mahavira’; as R. C. Guupta endowment lecture on ‘History of Mathematics’ in The 42nd Annual Conference of AMTI held at Warangal. After completing the translation work of Ganita Kaumudi of Narayana Pandita (1356AD) INTO Kannada, I wrote a separate book: Chandasastradinda Vikalpa Ganitada Vikasa; based on Ankapasa chapter of it in Kannada. Some excerpts of the book presented as a paper in the 5th Kannada Vijnana Sammelana held in September 2009 at Mangalore University. Subsequently while preparing for the lectures which were to be delivered at the ‘National Workshop on Ancient Indian Mathematics with special reference to Vedic Mathematics and Astronomy’ 20-24th September 2010 at Rashtriya Sanskrit Vidyapeetha Tirupathi. This book was compiled.

It is intended here to discuss combinatorics, i.e. permutations, combinations, partitions dealt with by Narayana Pandita in the Ankapasa chapter of his work Ganita kaumudi in the light and language of Sanskrit prosody.

The source material and preliminary aspects are mentioned briefly in ch.1 to 3. The basic principle involved in writing whole numbers using numeral notations (i.e., sankhya lekhana using anka-prastara)and prastara of varna vrttas is one and the same. This aspect is explicitly stated in ch.6 with the background of ch.4 and 5.

Galakriya pratyaya, which is more significant from mathematical point of view among sat-pratyayas, is dealt with for varnavrttas and permutations of digits in ch.7 and ch.8 respectively.

Figure of numbers formed on the basis of Pingala’s phorisms depicting the sarvaikadigalakriya has been named by his commentators Yadava and Hlayudha (10th century A.D.) as prasada prastara and Meru prastara respectively. Various numbers in the cells of each row of Meru-prastara gives the binomial coefficients in their proper sequence, and is based in the formula

C (n, r – 1) + C (n, r) = C (n + 1, r)

The figure of numbers containing binomial coefficients in its various forms has been given by Varahamihira (50 AD), Janasraya (prior to 600 A.D), Virahanka (between 600 -800 A.D), Jayadeva (between 600-900 A.D) and the author of Ratnamanjusa (c 900 A.D), Jayakirti (1000 A.D), Trivikrama and Hemacandra (c 1150 A.D). Bhattotpla (966 A.D) has explained Varahamihiras rule in detail.

Meru –prastara, the arithmetical triangle, consisting of binomial coefficients has often been known as Pascal’s-Triangle (1653 A.D), despite the fact that he was (by several centuries) not first to discuss it.

In Europe, at the outset of Renaissance, Newton (1642-1727 A.D) gave the binomial theorem for negative and fractional indices in 1676 A.D. subsequently, Leibnitz (1695 A.D), J Bernoulli and DeMoivres generalisedthe binomial theorem into polynomian theorem.

General formula to find the numbers in all the cells of (n+l)th row of Meru, i.e. mathematical formula for the number if ways of selecting out of ‘n’ objects ‘r’ objects at time, without regard for the order of selection;

 

Contents

 

     
1 Introduction 1-4
2 Vrttas and Jatis 4-5
3 Pratyayas 6-7
4 Pratyayas for varna-vrttas 8-22
  Prastara-Nastabheda-Uddistanka  
5 Vrna-Vrttas and permutation of digits 23-33
  Pratyayas for permutation of digits:  
  Prastara-Nastabedha-Uddistanka  
6 Principles underlying Narayana's rules for Prastara 34-38
  Pratyayas of varna-vrttas and serial arrangement of permutation of digits (ie. Sankhya lekhana and ankaprastara) Alternate rule suggested by the present author.  
7 Gala-kriya for varna-vrttas-Meru prasatara. 39-46
8 Meru-prastara-binomial coefficients. Permutation of digits in which a digit occurs 'r' times.  
9 Sankhya pratyaya for varna-vrttas. L 56-67
  Gunavargaja Tantra - sum of a GP.  
  Number kof Sama-Ardhasama-Visama vrttas  
10 Minor pratyayas Adhvayoga-Markati-Pataka 68-74
11 New pratyayas for permutation of digits and partition of a number. 75-77
12 Panktis (sequences), Merus (figures fo numbers) and Ladduka Karana (process of Ladduka) 78-111
13 Pratyayas for permutation of digits: 112-165
  (i) "n" different digits (or things) taken all at a time.  
  (ii) "n" digits (or things) taken all at a time when the digits are not different  
  (iii)"q" digits taken "p" of them at a time. (case akin to Varna-Vrttas)  
  (iv)Digits having a given sum  
  (v) Sankhya and other pratyayas for 'n' dissimilar digits taken 'r' at a time.  
14 Pratyayas for combination of 'n' dissimilar digits taking 'r' at a time. 166-189
15 Partition of numbers and Fibonacci series 190-191
16 Pratyayas for moric metre and partition of a number 192-218
  Prastara-Nastabedha-Uddistanka  
17 Ga-la-kriya for a moric metre. 219-241
  Special properties of the numbers in moric metre sequence.  
  Pratyayas for partitions of a number:  
  sankhya-avrti-antyabheda-urdhvapankti yoga. Sarva yoga-ankapata  
18 Summary and Epilogue 242-253
  Biblography 254

 

Sample Pages

















Development of Combinatiorics from The Pratyayas in Sanskrit Prosody

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About the Author

Sri Venugopal D. Heroor is an engineer by profession, and a keen and enthusiastic scholar of Ancient Indian Mathematics. He has brought out Five books on History of Mathematics and Mathematician of Ancient and Medieval India.

He has translated the Sanskrit works: Sridhara’s Patiganita and Trisatika, Ganitadhyaya of Brahmagupta’s Brahmasputa Siddhanta, Bhaskaracary’s jyotpatti and Lilavati, Chitrabhanu’sEkavimsati Prasnottara, Narayana Pandita’s Ganita Kaumudu in to Kannada.

He has translated many articles and research papers and has also Contributed original articles.in the present book, Development of Combinatorics from the Pratyayas in Sanskrit Prosody’the topic of combinatorics i.e., Permutations, combinations, partitions, sequence of binomial and polynomial coefficients, binomial theorem, figures containing these sequences and relations between them dealt with earlier by Narayana Pandita in his Ganita Kaumudi is discussed in the light of Sanskrit Prosody in modern terminology.

 

Foreward

Unique is the way of knowledge. Wonderful is the journey of learning. In the annals of Humanb history every branches of knowledge has its own pulsating twist and turns that link with so many other branches of wisdom. It is for this reason, our masters have made dictum.

He who is initiated in only one branch of learning will be incapable of saying anything decisive in the same field.

Likewise it is also said that some of the branches of knowledge become very significant and indispensable to so many other systems of learning. In the same lines we can also boldly proclaim that language and Mathematics form the bedrock of all knowledge, past and present.

Here logic is equated with mathematics and this is understood by every serious student of it. Mathematics will never provide us with any ‘information’. It can only process it and bring out the details of the same in many ways that show wonderful internal relationships and equations. Likewise, language too, having the indispensable gramatic governance represents the needed information. But by itself, it cannot be that. However, thepath of mathematics I ssymbols while that of the language is sounds. But we do have the inseparable relationship between signs and sounds. In this way, the sequence of cause and effect, well found in language / grammar joins its hands with the same chain found in the counts and concepts of mathematics / logic.

Prosody is the science of metres that are employed in the art of versification. This essentially deals with the pattern of longer and shorter sounds /stressed and unstressed syllables. (similar to, but not essentially the same).In India we have a Great heritage for this science as it is one among the six limbs o f the Vedas. Since time immemorial, we had many verities of metres. In the vedic texts themselves we find several scores of metres which have their basic patterns in the famed seven groups. Coming to the classical period, the number of metres goes to several hundreds and the masters in this field have done a laudable job by giving beautiful classification and analysis. In this period perhaps, we see a great deal of work done on the mathematical relationship revealed in the patterns of longer and shorter sounds called as gurus and laghus – found in the matrix of any metre.

Here we should note that the three main divisions of Indian/Sanskrit Metres viz. the Varna Vrttas, the Matrajati-s and the Aksharajati-s / Visayajati-s have their own unique challenges to mathemaics and all these are will answered by the scholars in the respective fields. Thus in the finest art of poetry, Mathematics, the queen among sciences has made her way. But this has to do nothing with the aesthetics of prosody. Yet, we have to bear in the mind that the prosodic patterns alone create aesthetic effect and such a pattern has an inbuilt logic. However, these primary constructions are not engineered. They are essentially creations which sprang spontaneously. But again, we gave to say that such a spontaneous aesthetic creation has an inbuilt wisdom and awareness. Thus the whole issue leads us towards the inseparability of head and heart at the root. However, we are not going to those details here.

Our prosodists starting from Pingala, have given six folded scheme called satpratyaya prakriya to know some of the essential and mathematically tenable things in metrics. These pratyayas, called so because they reveal such details of prosody help us in presenting the possible metric patterns of guru laghus in one main set, search for us a lost pattern, locate a specific pattern, bring out the details of various metres, make us know the total number of verities in a set without going to the laborious process of prastara and hint at the possible space needed for a full-filedged presentation of the matrix. Among these, ekadvyadilagakriya is the most important and has many ways of accomplishing the same. Thus Mathematicians like Bhaskaracharya too have brought out their respective versions.

In this light, the present work of my esteemed learned friend Sri Venugopal D. Heroor has to be understood. Mr. Heroor has spared no pains in making his work, the ‘Development of Combinatorics from the pratyays in Sanskrit prosody’ both exhaustive and comprehensive. His literature survey is thorough and exposition is lucid. Since long, he has been in the field of Indian mathematics, which is indeed a hard nut to crack as it demands the knowledge of both logic and language! However, Mr. Heroor’s many earlier publications have carved a niche of his aim and this publication is going to become yet another feather in his cap. Therefore I wholeheartedly congratulate him and strongly recommend his work to the lare of Indic learning.

 

Preface

My study on the topic of ‘Combinatorics in Prosody’ started with the article ‘Use of Permutations and Combination in India’ by B. Datta and A. N. Singh, which I have translated into Kannada and published in the Kannada monthly UTTHANA, in June 2002. Later in December 2007, I gave a lecture on ‘The Concept of Binary place value system of numeration in ‘sat-pratyaya sutras of Mahavira’; as R. C. Guupta endowment lecture on ‘History of Mathematics’ in The 42nd Annual Conference of AMTI held at Warangal. After completing the translation work of Ganita Kaumudi of Narayana Pandita (1356AD) INTO Kannada, I wrote a separate book: Chandasastradinda Vikalpa Ganitada Vikasa; based on Ankapasa chapter of it in Kannada. Some excerpts of the book presented as a paper in the 5th Kannada Vijnana Sammelana held in September 2009 at Mangalore University. Subsequently while preparing for the lectures which were to be delivered at the ‘National Workshop on Ancient Indian Mathematics with special reference to Vedic Mathematics and Astronomy’ 20-24th September 2010 at Rashtriya Sanskrit Vidyapeetha Tirupathi. This book was compiled.

It is intended here to discuss combinatorics, i.e. permutations, combinations, partitions dealt with by Narayana Pandita in the Ankapasa chapter of his work Ganita kaumudi in the light and language of Sanskrit prosody.

The source material and preliminary aspects are mentioned briefly in ch.1 to 3. The basic principle involved in writing whole numbers using numeral notations (i.e., sankhya lekhana using anka-prastara)and prastara of varna vrttas is one and the same. This aspect is explicitly stated in ch.6 with the background of ch.4 and 5.

Galakriya pratyaya, which is more significant from mathematical point of view among sat-pratyayas, is dealt with for varnavrttas and permutations of digits in ch.7 and ch.8 respectively.

Figure of numbers formed on the basis of Pingala’s phorisms depicting the sarvaikadigalakriya has been named by his commentators Yadava and Hlayudha (10th century A.D.) as prasada prastara and Meru prastara respectively. Various numbers in the cells of each row of Meru-prastara gives the binomial coefficients in their proper sequence, and is based in the formula

C (n, r – 1) + C (n, r) = C (n + 1, r)

The figure of numbers containing binomial coefficients in its various forms has been given by Varahamihira (50 AD), Janasraya (prior to 600 A.D), Virahanka (between 600 -800 A.D), Jayadeva (between 600-900 A.D) and the author of Ratnamanjusa (c 900 A.D), Jayakirti (1000 A.D), Trivikrama and Hemacandra (c 1150 A.D). Bhattotpla (966 A.D) has explained Varahamihiras rule in detail.

Meru –prastara, the arithmetical triangle, consisting of binomial coefficients has often been known as Pascal’s-Triangle (1653 A.D), despite the fact that he was (by several centuries) not first to discuss it.

In Europe, at the outset of Renaissance, Newton (1642-1727 A.D) gave the binomial theorem for negative and fractional indices in 1676 A.D. subsequently, Leibnitz (1695 A.D), J Bernoulli and DeMoivres generalisedthe binomial theorem into polynomian theorem.

General formula to find the numbers in all the cells of (n+l)th row of Meru, i.e. mathematical formula for the number if ways of selecting out of ‘n’ objects ‘r’ objects at time, without regard for the order of selection;

 

Contents

 

     
1 Introduction 1-4
2 Vrttas and Jatis 4-5
3 Pratyayas 6-7
4 Pratyayas for varna-vrttas 8-22
  Prastara-Nastabheda-Uddistanka  
5 Vrna-Vrttas and permutation of digits 23-33
  Pratyayas for permutation of digits:  
  Prastara-Nastabedha-Uddistanka  
6 Principles underlying Narayana's rules for Prastara 34-38
  Pratyayas of varna-vrttas and serial arrangement of permutation of digits (ie. Sankhya lekhana and ankaprastara) Alternate rule suggested by the present author.  
7 Gala-kriya for varna-vrttas-Meru prasatara. 39-46
8 Meru-prastara-binomial coefficients. Permutation of digits in which a digit occurs 'r' times.  
9 Sankhya pratyaya for varna-vrttas. L 56-67
  Gunavargaja Tantra - sum of a GP.  
  Number kof Sama-Ardhasama-Visama vrttas  
10 Minor pratyayas Adhvayoga-Markati-Pataka 68-74
11 New pratyayas for permutation of digits and partition of a number. 75-77
12 Panktis (sequences), Merus (figures fo numbers) and Ladduka Karana (process of Ladduka) 78-111
13 Pratyayas for permutation of digits: 112-165
  (i) "n" different digits (or things) taken all at a time.  
  (ii) "n" digits (or things) taken all at a time when the digits are not different  
  (iii)"q" digits taken "p" of them at a time. (case akin to Varna-Vrttas)  
  (iv)Digits having a given sum  
  (v) Sankhya and other pratyayas for 'n' dissimilar digits taken 'r' at a time.  
14 Pratyayas for combination of 'n' dissimilar digits taking 'r' at a time. 166-189
15 Partition of numbers and Fibonacci series 190-191
16 Pratyayas for moric metre and partition of a number 192-218
  Prastara-Nastabedha-Uddistanka  
17 Ga-la-kriya for a moric metre. 219-241
  Special properties of the numbers in moric metre sequence.  
  Pratyayas for partitions of a number:  
  sankhya-avrti-antyabheda-urdhvapankti yoga. Sarva yoga-ankapata  
18 Summary and Epilogue 242-253
  Biblography 254

 

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