It is now being increasingly recognized that ancient Sanskrit Literature contains profound wisdom of physical and social sciences. The scientific knowledge base presented by our Rishis and perceptive thinkers covers a wide canvas of subjects such as Physics, Chemistry, Mathematics, Astronomy, Metallurgy, BioTechnology and Environment Science etc.
Unfortunately due to a very long period of intellectual domination of the forces from outside the region, we have lost our awareness about the profound contributions made by our ancestors that are hidden in the vast Sanskrit literature. Our continued apathy towards Sanskrit language over the recent centuries has also contributed to our lack of awareness about rich scientific heritage. Our ignorance about the richness of our heritage has made us to lose our sense of pride for what is ours. As a result of all this, our sense of confidence in our own capabilities has also been shaken.
Knowledge of Sanskrit and perceptive analysis of contents of our ancient Sanskrit literature would be of great help in getting a good insight into our rich scientific heritage. It is the responsibility of Sanskrit institutions to arouse the interests of the modern world in Sanskrit by unraveling the profound scientific wisdom that is contained in the Sanskrit literature and by demonstrating its relevance to the modern world. It is only when for dealing with the contemporary challenges of life would be a link between the Sanskrit world and the modern world is established that the acceptability of Sanskrit as a relevant subject for dealing with the contemporary challenges of life would be enhanced. It is with this object in view that the Rashtriya Sanskrit Vidyapeetha, Tirupati, launched a series of publication of small monographs which cover wide convass of subjects such as Physics, Chemistry, Astronomy, Ayurveda and Mathematics etc. for the enlightenment of younger generations of India.
Prof. Venkatesha Murthy, who is a profound scholar of Mathematics and a member of our SanskritScience Study Centre, has prepared this monograph with his vast experience of teaching ‘Indian Mathematics’ and conducting seminars, workshops and exhibitions related to the subject. I congratulate him for preparing a wonderful monograph.
I hope that this monograph would be found useful by students and scholars alike and inspire many students of Mathematics.
We feel immensely happy and proud to present this monograph titled “Indian Mathematics in Sanskrit: Concepts and Achievements” to the young readers who have a longing to have a glimpse of the wisdom of ancient India.
There is a general belief that contribution of ancient India is limited to the field of human particularly that of religion and philosophy. Of course, no other civilization can claim superiority over what India has achieved in that area. Still if we go through the pages of this monograph, we will realize that in the area of Mathematics India’s contribution was amazing. Now India has already lost much of its ancient lore due to the impact of foreign invasion. What remains also is largely unknown as most of it is in Sanskrit. Hence, we feel that there is an urgent need to save the remaining knowledge and build up awareness in all concerned, especially among younger generations who are not aware of our ancient scientific heritage contained in Sanskrit.
Shouldering the responsibility of creating interest in Sanskrit and work on scientific heritage written in Sanskrit among younger generations and exposing their relevance to the modern world, Rashtriya Sanskrit Vidyapeetha took the initiative of organizing Sanskrit Science exhibitions, publishing and propagating the Scientific literature of Sanskrit through out India.
I should mention at the out set that the inspiration to publish such monographs came for our Hon’ble Chancellor Dr. V.R. Panchamukhi, Economist of International fame, and versatile scholar in Sanskrit.
I would like to express my sincere thanks to Prof. Venkatesha Murthy for preparing this wonderful monograph.
If this small monograph kindles a tiny flame of desire in the hearts of young readers for studying Sanskrit, the store house of past glory of India., we feel rewarded for our efforts.
“The history of the development of Mathematics in India is as old as the civilization of its people itself. It begins with the rudiments of metrology and computations in prehistoric times, of which some fragmentary evidence has survived to this day. The sacred literature of the Vedic Hindus – The Samhitas, the Kalpasutras and the Vedangas – contain enough materials, albeit scattered, to help form a good idea of the Mathematical ability during the time of development of this class of literature. The Sulbasutras which form a part of the Kalpasutras are a veritable storehouse of information concerning enumeration, arithmetical operations, fractions, properties of rectilinear figures, the socalled Pythagoras Theorem, surds, irrational numbers, quadratic and indeterminate equations and related matters.
The world sulba (Sulva) means a ‘cord’ a ‘rope’ or a ‘string’, and its sulb signifies ‘measuring’ or ‘act of measurement’. Therefore, sulbasutras are a collection of rules concerning measurements with the help of a cord of various linear, spatial or threedimensional figures. Sulbasutras deal with rules for the measurements and constructions of various sacrificial alters and consequently involve geometrical propositions and problems relating to rectilinear figures, their combinations and transformations, squaring the circle, circling the square as well as arithmetical and algebraic solutions of problems arising out of such measurements and constructions. The Baudhayana Sulbasutra is the oldest (600500 B.C.). The Manava Sulbasutra is posterior to Baudhayana and contains descriptions of a number of altars, not found in other works. The Apastamba Sulbasutra (500400 B.C.) gives the same rules as were found in Baudhayana Sulbasutra. The Katyayana Sulbasutra (400300 B.C.) is more succinct and more systematic.
Of the various sects that attained prominence in the closing phase of the (active) Vedic period, the Jainas deserve special notice for their interest in, and cultivation of, Mathematics. Their canonical literature lays great emphasis on Mathematics and enumerates various topics such as number reckoning, fundamental operations, geometry, mensuration, fractions, equations, permutations and combinations.
The medieval period witnessed growth of a sizeable mathematical literature in Arabic and Persian and presented as opportunity for crossfertilization of the efforts of two distinct cultures. Records of late ancient and medieval times show that Indian Mathematics stimulated mathematical endeavours abroad and itself received inspiration from neighbouring and distant cultural areas. Such stimuli have been noticed even in the case of the complex and manysided phenomenon of European Renaissance of the sixteenth century.”
[“A Concise History of Science in India” – (Editors) D.M. Bose, S.N. Sen, B.V. Subbarayappa, INSA, New Delhi, (1989) p. 136]
The above passages is the main source of inspiration to prepare charts for Sanskrit Science Exhibition being organized all over India by Rashtriya Sanskrit Vidyapeetha, (Deemed University), Tirupati. The project of preparing these charts on “Indian Mathematics in Sanskrit: Concepts and Achievements” as a part of “Sanskrit – Science Exhibition” undertaken at the suggestion of Dr. V.R. Panchamukhi, Chancellor, Rashtriya Sanskrit Vidyapeetha (Deemed University), Tirupati during its Annual Convocation – 2000. Later, these exhibits were exhibited in several Universities, Centres of Higher learning, and also during several special occasions all over India. Recently, “Sanskrit – Science Exhibition and Seminar on ‘Relevance of Ancient Scientific Concepts to the Present day’ “was held at St. Petersburg (Russia), on invitation from Government of Russia. The programme at Russia was sponsored by the Directorate of Science and Technology Government of India. Lecture was also delivered with powerpoint presentations during these occasions.
It gives me immense in acknowledging my sincere gratitude to the respected Chancellor Dr. V.R. Panchamukhi, Prof. K.E. Govindan, Vicechancellor i/c, Prof. D. Prahlada Char, Former ViceChancellor, Rashtriya Sanskrit Vidyapeetha, Tirupati, Prof. V.B. Subbarayappa, Honorary Professor, National Institute of Advanced Studies, and Chief of Editorial Board, SanskritScience Study Centre, Rashtriya Sanskrit Vidyapeetha, Tirupati, and my dear friend Prof. V. Muralidhara Sharma, Coordinator, Sanskrit – Science Study Centre for bestowing these rare opportunities. My sincere thanks to Sri Chenraj Jain, Chairman, Jains Group of Institutions and Chancellor, MATS University and to Dr. Gruuraja Karajagi, eminent educationist and the Director, International Academy for Creative Teaching, Bangalore for their support and encouragement for all my endeavours.
Foreword  I  
Preface  II  
Contents  III  

1  43  
1.  Introduction: Sulbasutra Theorem (Pythagorean Theorem) and its application  3 to 7 
2.  Theorem: Square on the diagonal of a rectangle  Chart  4 
3.  Theorem from Manava Sulbasutra (Pythagoras theorem)  Chart  6 
4.  Theorem: Square on the diagonal of a square from Baudhayana Sulbasutra –Chart  8 
5.  Surds and Their Approximate Values from Katyayana Sulvasutra  9 
6.  Value of Square root of 2 from Baudhayana sulvasutra  Chart  10 
7.  Verification of Sulbasutra value of Square of 2 – Thibaut’s and Rodet’s approximations  11 to 14 
8.  An approximation to square root of 3  15 
9.  To get Ö3 Geometrically – Baudhayana Sulbasutra  Chart  16 
10.  (Ö3 and 1/Ö3) in Katyayana Sulvasutra  Chart  17 & 18 
11.  Ratio of circumference of a circle to its diameter from Sulbasutras  19 
12.  Geometrical basis for Aryabhata’s Value of ‘Ratio of circumference of a circle to its diameter’  Chart  20 
13.  Algorithm to find Aryabhata’s Value of ‘Ratio of circumference of a circle to its diameter’  21 
14.  Algorithm to Double the Number of Sides ‘n’ of an Inscribed Regular Polygon – Chart & Explanation  22 & 23 
15.  Algorithm to find a Value of p of Aryabhata – I (c. A.D. 476) – Chart & Explanation  24 & 25 
16.  Sloka from Aryabhatiya (c. A.D. 499) about Nearly Accurate value of p  Chart  26 
17.  Aryabhata – I, and Others (c. A.D. 476) about Nearly Accurate value of p.  Explanation  27 
18.  Power Series of p Madhava and of Gottgried Wilhelm Leibnitz – Chart & Explanation  28 & 29 
19.  Definition of Indian Trigonometrical Functions – Chart & Explanation  30 & 31 
20.  Origin of the term Sine and other Trigonometrical terms – Chart & Explanation  32 & 33 
21.  Algotrithm to find desired number of Rsines  Aryabhatiya  34 & 35 
22.  A General Method to Calculate (‘N’, the Desired number of) Rsines – Chart & Explanation  36 & 37 
23.  To Calculate Six Rsines Geometrically – Charts & Explanation  38 & 41 
24.  Value of a Radian Measure in Aryabhatiya – Chart & Explanation  42 & 43 

45 to 51  
1.  Etymology of the word Zero and Operations with Zero – Chart & Explanation  46 & 47 
2.  Zero in Lilavati and Bijaganita of BhaskaracaryaII (b. A.D. 1114) – Chart & Explanation  48 & 49 
3.  Universal property of infinity in Bijaganita of Bhascara II – Chart & Explanation  50 & 51 

53 to 63  
1.  Numbers and Numerals in Sanskrit works  55 
2.  Sanskrit Words as Numerals – Chart & Explanation  56 & 57 
3.  Sanskrit Alphabets as Numerals; Sloka and Explanation  58 & 59 
4.  A Ready Reckoner  60 
5.  and Buddhist Alchemist Nagarjuna’s Magic Square – Charts & Explanation  61 to 63 

65 to 79  
1.  Aryabhatiya Numeral System of Aryabhata 1  67 
2.  Sloka from Aryabhatiya and a Ready Reckoner for Aryabhatiya Numeral System  68 & 69 
3.  Sloka from Aryabhatiya on the Aryabhatiya Numerals denoting number of revolutions of Geocentric planets in a yuga (43,20,000 Years)  70 
4.  Table showing the order of Geocentric planets in the increasing order of their number of revolutions in a yuga in the Aryabhatiya Numerals and International numerals  71 
5.  Explanation of converting Aryabhatiya Numerals denoting number of revolutions of Geocentric planets in a yuga (43,20,000 Years) into International Numerals.  72 to 78 
6.  Present day sidereal periods of GeoCentric planets from Science Data Sources and from Aryabhatiya (A.D. 499) of Aryabhata –1 :A Comparison  79 

81 to 90  
1.  Why Weekdays are Named so? – A sloka Aryabhatiya and an Overview  82 & 83 
2.  Explanation for Naming the Weekdays a table better understanding  84 to 85 
3.  A sloka on the order of Names of Weekdays in Atharvavedajyotisha  86 
4.  Comments on the Sloka on Weekdays in Aryabhatiya  87 
5.  A Working Model to show why weekdays are named so – Chart & Explanation  88 & 89 
6.  A Static Model to get the names of Geocentric planets from the names of weekdays Explanation & Chart  89 & 90 

91 to 122  
1.  Need for Rational Approximation of ÖN  93 
2.  Vargaprakruti: Explanation of the equation Nx2 + h = y2  95 to 97 
3.  Bhavana of Brahmagupta.  99 to 105 
(i) Rule 1 & Rule 2 of Bhavana of Brahmagupta.  100 to 103  
(ii) Supplementary Rule with Examples  104 & 105  
4.  Chakravala  106 to 122 
(i) Differentiating Bhavana and Cakravala  106  
(ii) Cakravala of Acarya Jayadeva  107 to 109  
(iii) Cakravala of BhaskaraII  110 & 111  
(iv) Cakravala – an Extension of Bhavana  112 & 113  
(v) Speciality in Cakravala of Acarya Jayadeva  114 & 115  
(vi) To find least integral values of 61x2 + 1 = y2 using Jayadeva – Bhaskara method  116 & 117  
(vii) To find least integral values of 61x2 + 1 = y2 using Language’s Method  118 & 119  
(viii) General Method of Cakravala to solve Nx2 + 1 = y2 for integral values of x and y and 67x2 + 1 = y2  120 to 122  
5.  Epilogue  123 to 125 
References:  127 & 128 
It is now being increasingly recognized that ancient Sanskrit Literature contains profound wisdom of physical and social sciences. The scientific knowledge base presented by our Rishis and perceptive thinkers covers a wide canvas of subjects such as Physics, Chemistry, Mathematics, Astronomy, Metallurgy, BioTechnology and Environment Science etc.
Unfortunately due to a very long period of intellectual domination of the forces from outside the region, we have lost our awareness about the profound contributions made by our ancestors that are hidden in the vast Sanskrit literature. Our continued apathy towards Sanskrit language over the recent centuries has also contributed to our lack of awareness about rich scientific heritage. Our ignorance about the richness of our heritage has made us to lose our sense of pride for what is ours. As a result of all this, our sense of confidence in our own capabilities has also been shaken.
Knowledge of Sanskrit and perceptive analysis of contents of our ancient Sanskrit literature would be of great help in getting a good insight into our rich scientific heritage. It is the responsibility of Sanskrit institutions to arouse the interests of the modern world in Sanskrit by unraveling the profound scientific wisdom that is contained in the Sanskrit literature and by demonstrating its relevance to the modern world. It is only when for dealing with the contemporary challenges of life would be a link between the Sanskrit world and the modern world is established that the acceptability of Sanskrit as a relevant subject for dealing with the contemporary challenges of life would be enhanced. It is with this object in view that the Rashtriya Sanskrit Vidyapeetha, Tirupati, launched a series of publication of small monographs which cover wide convass of subjects such as Physics, Chemistry, Astronomy, Ayurveda and Mathematics etc. for the enlightenment of younger generations of India.
Prof. Venkatesha Murthy, who is a profound scholar of Mathematics and a member of our SanskritScience Study Centre, has prepared this monograph with his vast experience of teaching ‘Indian Mathematics’ and conducting seminars, workshops and exhibitions related to the subject. I congratulate him for preparing a wonderful monograph.
I hope that this monograph would be found useful by students and scholars alike and inspire many students of Mathematics.
We feel immensely happy and proud to present this monograph titled “Indian Mathematics in Sanskrit: Concepts and Achievements” to the young readers who have a longing to have a glimpse of the wisdom of ancient India.
There is a general belief that contribution of ancient India is limited to the field of human particularly that of religion and philosophy. Of course, no other civilization can claim superiority over what India has achieved in that area. Still if we go through the pages of this monograph, we will realize that in the area of Mathematics India’s contribution was amazing. Now India has already lost much of its ancient lore due to the impact of foreign invasion. What remains also is largely unknown as most of it is in Sanskrit. Hence, we feel that there is an urgent need to save the remaining knowledge and build up awareness in all concerned, especially among younger generations who are not aware of our ancient scientific heritage contained in Sanskrit.
Shouldering the responsibility of creating interest in Sanskrit and work on scientific heritage written in Sanskrit among younger generations and exposing their relevance to the modern world, Rashtriya Sanskrit Vidyapeetha took the initiative of organizing Sanskrit Science exhibitions, publishing and propagating the Scientific literature of Sanskrit through out India.
I should mention at the out set that the inspiration to publish such monographs came for our Hon’ble Chancellor Dr. V.R. Panchamukhi, Economist of International fame, and versatile scholar in Sanskrit.
I would like to express my sincere thanks to Prof. Venkatesha Murthy for preparing this wonderful monograph.
If this small monograph kindles a tiny flame of desire in the hearts of young readers for studying Sanskrit, the store house of past glory of India., we feel rewarded for our efforts.
“The history of the development of Mathematics in India is as old as the civilization of its people itself. It begins with the rudiments of metrology and computations in prehistoric times, of which some fragmentary evidence has survived to this day. The sacred literature of the Vedic Hindus – The Samhitas, the Kalpasutras and the Vedangas – contain enough materials, albeit scattered, to help form a good idea of the Mathematical ability during the time of development of this class of literature. The Sulbasutras which form a part of the Kalpasutras are a veritable storehouse of information concerning enumeration, arithmetical operations, fractions, properties of rectilinear figures, the socalled Pythagoras Theorem, surds, irrational numbers, quadratic and indeterminate equations and related matters.
The world sulba (Sulva) means a ‘cord’ a ‘rope’ or a ‘string’, and its sulb signifies ‘measuring’ or ‘act of measurement’. Therefore, sulbasutras are a collection of rules concerning measurements with the help of a cord of various linear, spatial or threedimensional figures. Sulbasutras deal with rules for the measurements and constructions of various sacrificial alters and consequently involve geometrical propositions and problems relating to rectilinear figures, their combinations and transformations, squaring the circle, circling the square as well as arithmetical and algebraic solutions of problems arising out of such measurements and constructions. The Baudhayana Sulbasutra is the oldest (600500 B.C.). The Manava Sulbasutra is posterior to Baudhayana and contains descriptions of a number of altars, not found in other works. The Apastamba Sulbasutra (500400 B.C.) gives the same rules as were found in Baudhayana Sulbasutra. The Katyayana Sulbasutra (400300 B.C.) is more succinct and more systematic.
Of the various sects that attained prominence in the closing phase of the (active) Vedic period, the Jainas deserve special notice for their interest in, and cultivation of, Mathematics. Their canonical literature lays great emphasis on Mathematics and enumerates various topics such as number reckoning, fundamental operations, geometry, mensuration, fractions, equations, permutations and combinations.
The medieval period witnessed growth of a sizeable mathematical literature in Arabic and Persian and presented as opportunity for crossfertilization of the efforts of two distinct cultures. Records of late ancient and medieval times show that Indian Mathematics stimulated mathematical endeavours abroad and itself received inspiration from neighbouring and distant cultural areas. Such stimuli have been noticed even in the case of the complex and manysided phenomenon of European Renaissance of the sixteenth century.”
[“A Concise History of Science in India” – (Editors) D.M. Bose, S.N. Sen, B.V. Subbarayappa, INSA, New Delhi, (1989) p. 136]
The above passages is the main source of inspiration to prepare charts for Sanskrit Science Exhibition being organized all over India by Rashtriya Sanskrit Vidyapeetha, (Deemed University), Tirupati. The project of preparing these charts on “Indian Mathematics in Sanskrit: Concepts and Achievements” as a part of “Sanskrit – Science Exhibition” undertaken at the suggestion of Dr. V.R. Panchamukhi, Chancellor, Rashtriya Sanskrit Vidyapeetha (Deemed University), Tirupati during its Annual Convocation – 2000. Later, these exhibits were exhibited in several Universities, Centres of Higher learning, and also during several special occasions all over India. Recently, “Sanskrit – Science Exhibition and Seminar on ‘Relevance of Ancient Scientific Concepts to the Present day’ “was held at St. Petersburg (Russia), on invitation from Government of Russia. The programme at Russia was sponsored by the Directorate of Science and Technology Government of India. Lecture was also delivered with powerpoint presentations during these occasions.
It gives me immense in acknowledging my sincere gratitude to the respected Chancellor Dr. V.R. Panchamukhi, Prof. K.E. Govindan, Vicechancellor i/c, Prof. D. Prahlada Char, Former ViceChancellor, Rashtriya Sanskrit Vidyapeetha, Tirupati, Prof. V.B. Subbarayappa, Honorary Professor, National Institute of Advanced Studies, and Chief of Editorial Board, SanskritScience Study Centre, Rashtriya Sanskrit Vidyapeetha, Tirupati, and my dear friend Prof. V. Muralidhara Sharma, Coordinator, Sanskrit – Science Study Centre for bestowing these rare opportunities. My sincere thanks to Sri Chenraj Jain, Chairman, Jains Group of Institutions and Chancellor, MATS University and to Dr. Gruuraja Karajagi, eminent educationist and the Director, International Academy for Creative Teaching, Bangalore for their support and encouragement for all my endeavours.
Foreword  I  
Preface  II  
Contents  III  

1  43  
1.  Introduction: Sulbasutra Theorem (Pythagorean Theorem) and its application  3 to 7 
2.  Theorem: Square on the diagonal of a rectangle  Chart  4 
3.  Theorem from Manava Sulbasutra (Pythagoras theorem)  Chart  6 
4.  Theorem: Square on the diagonal of a square from Baudhayana Sulbasutra –Chart  8 
5.  Surds and Their Approximate Values from Katyayana Sulvasutra  9 
6.  Value of Square root of 2 from Baudhayana sulvasutra  Chart  10 
7.  Verification of Sulbasutra value of Square of 2 – Thibaut’s and Rodet’s approximations  11 to 14 
8.  An approximation to square root of 3  15 
9.  To get Ö3 Geometrically – Baudhayana Sulbasutra  Chart  16 
10.  (Ö3 and 1/Ö3) in Katyayana Sulvasutra  Chart  17 & 18 
11.  Ratio of circumference of a circle to its diameter from Sulbasutras  19 
12.  Geometrical basis for Aryabhata’s Value of ‘Ratio of circumference of a circle to its diameter’  Chart  20 
13.  Algorithm to find Aryabhata’s Value of ‘Ratio of circumference of a circle to its diameter’  21 
14.  Algorithm to Double the Number of Sides ‘n’ of an Inscribed Regular Polygon – Chart & Explanation  22 & 23 
15.  Algorithm to find a Value of p of Aryabhata – I (c. A.D. 476) – Chart & Explanation  24 & 25 
16.  Sloka from Aryabhatiya (c. A.D. 499) about Nearly Accurate value of p  Chart  26 
17.  Aryabhata – I, and Others (c. A.D. 476) about Nearly Accurate value of p.  Explanation  27 
18.  Power Series of p Madhava and of Gottgried Wilhelm Leibnitz – Chart & Explanation  28 & 29 
19.  Definition of Indian Trigonometrical Functions – Chart & Explanation  30 & 31 
20.  Origin of the term Sine and other Trigonometrical terms – Chart & Explanation  32 & 33 
21.  Algotrithm to find desired number of Rsines  Aryabhatiya  34 & 35 
22.  A General Method to Calculate (‘N’, the Desired number of) Rsines – Chart & Explanation  36 & 37 
23.  To Calculate Six Rsines Geometrically – Charts & Explanation  38 & 41 
24.  Value of a Radian Measure in Aryabhatiya – Chart & Explanation  42 & 43 

45 to 51  
1.  Etymology of the word Zero and Operations with Zero – Chart & Explanation  46 & 47 
2.  Zero in Lilavati and Bijaganita of BhaskaracaryaII (b. A.D. 1114) – Chart & Explanation  48 & 49 
3.  Universal property of infinity in Bijaganita of Bhascara II – Chart & Explanation  50 & 51 

53 to 63  
1.  Numbers and Numerals in Sanskrit works  55 
2.  Sanskrit Words as Numerals – Chart & Explanation  56 & 57 
3.  Sanskrit Alphabets as Numerals; Sloka and Explanation  58 & 59 
4.  A Ready Reckoner  60 
5.  and Buddhist Alchemist Nagarjuna’s Magic Square – Charts & Explanation  61 to 63 

65 to 79  
1.  Aryabhatiya Numeral System of Aryabhata 1  67 
2.  Sloka from Aryabhatiya and a Ready Reckoner for Aryabhatiya Numeral System  68 & 69 
3.  Sloka from Aryabhatiya on the Aryabhatiya Numerals denoting number of revolutions of Geocentric planets in a yuga (43,20,000 Years)  70 
4.  Table showing the order of Geocentric planets in the increasing order of their number of revolutions in a yuga in the Aryabhatiya Numerals and International numerals  71 
5.  Explanation of converting Aryabhatiya Numerals denoting number of revolutions of Geocentric planets in a yuga (43,20,000 Years) into International Numerals.  72 to 78 
6.  Present day sidereal periods of GeoCentric planets from Science Data Sources and from Aryabhatiya (A.D. 499) of Aryabhata –1 :A Comparison  79 

81 to 90  
1.  Why Weekdays are Named so? – A sloka Aryabhatiya and an Overview  82 & 83 
2.  Explanation for Naming the Weekdays a table better understanding  84 to 85 
3.  A sloka on the order of Names of Weekdays in Atharvavedajyotisha  86 
4.  Comments on the Sloka on Weekdays in Aryabhatiya  87 
5.  A Working Model to show why weekdays are named so – Chart & Explanation  88 & 89 
6.  A Static Model to get the names of Geocentric planets from the names of weekdays Explanation & Chart  89 & 90 

91 to 122  
1.  Need for Rational Approximation of ÖN  93 
2.  Vargaprakruti: Explanation of the equation Nx2 + h = y2  95 to 97 
3.  Bhavana of Brahmagupta.  99 to 105 
(i) Rule 1 & Rule 2 of Bhavana of Brahmagupta.  100 to 103  
(ii) Supplementary Rule with Examples  104 & 105  
4.  Chakravala  106 to 122 
(i) Differentiating Bhavana and Cakravala  106  
(ii) Cakravala of Acarya Jayadeva  107 to 109  
(iii) Cakravala of BhaskaraII  110 & 111  
(iv) Cakravala – an Extension of Bhavana  112 & 113  
(v) Speciality in Cakravala of Acarya Jayadeva  114 & 115  
(vi) To find least integral values of 61x2 + 1 = y2 using Jayadeva – Bhaskara method  116 & 117  
(vii) To find least integral values of 61x2 + 1 = y2 using Language’s Method  118 & 119  
(viii) General Method of Cakravala to solve Nx2 + 1 = y2 for integral values of x and y and 67x2 + 1 = y2  120 to 122  
5.  Epilogue  123 to 125 
References:  127 & 128 