Dr. Ramamurthy, has been in the forefront of imparting knowledge based studies in the areas of Vedas, Samskrit, Banking, related IT solutions, Information Security, IT Audit and so on. His thirst for continuous learning does not subside. He did research on an unique topic “Information Technology and Samskrit” and obtained Ph.D. – doctorate degree from University of Madras even at the age of late fifties. He has so far penned 18 books on religious literature, Software Testing and Banking topics. Currently he is working on a project of developing a Samskrit based compiler.
Foreword
To text much 'mathema' in a little, multum in parvo, sleep is implicated: by derivation, jagrat is adverted. With a primer of an exclusive kind to which this work belongs, from the post-modern age of hypnoglyph, may the innocents in many of us wake up into a realm of a larger Time vectored 'Vedic', with modalities of dormancy impounded in it by the superstructures of the Western, compliant or not!
If all knowledge be acknowledgement, as Plato would have it, it is good to be ekphrastic like Mr. N. Ramamurthy, by indirection, often tabling that number is neither revelry nor devilry, or geometry is none of the debauchery or carousal of Space. Mere mention of Satapatabrahmana and such and such is gnomic to suggest that when our sleep is profound, we stand and are put in hold! Hu'pnos hedraios, orthoi, nustagmo's
[Epidemics 6.4.15; Littre':310].
My purport is to declaim, though it might seem a bit bizarre, by an analogy: to use the surmise of Galen (129-199 C.E), the late Classical successor of Hippocrates, and the so called advanced most wakeful number-theoreticians of our day may achieve a soothing slumber if they remain erect and vigilant until they begin to nod and wink!
A saying goes: Even Homer nods! If we return to origins, in peripety, we experience a jolt or an un-hinged-ness of our advanced-ness: a constraint to sleepwalk, awakening only when stumble on a stone, paleolithic! I mean, tit-bits are misnomers for even if they pass for so so, cover they do an aura of a decisive juncture of an essential moment, the author attempts to recreate, tending to be minimally narrational. Of course, Laplace supports the choice of Mr. Ramamurthy's locution 'pragnantesten', most pregnant, for a total emergence of mathema's progeny from parent Vedic, with a historic-Indic sense and demystification of not just instants but incidents, relating ensembles of Time and Space. The tit-bit type of primering the profound is endemic to Mr. Ramamurthy's, for his is spiring fulfilled continuity of Shree Chakra resolution.
The contents of this book are a phalanx in an under-coding for a spirited adoption of a subtle differential from our era's norms. Come what may, if a lay starter is schooled in these hundred and odd pages of ages of labor, the durable circadian boundaries blur. Mathematical logic and mathematical rhetoric as more than mere approval of a diaristic discourse must help the learner thank the impersonal chronicle of the theological track of a believer's subjectivity, engendering no muddle in the middle we owe to!
Ramamurthy in his more-than-a-text-book seems to postulate a categorical neither-here-nor-there-ness of what the student world of mathematics has been in. Any way the book is not to be viewed as a recordation calendrically, though there is an overtone of temporal character in the signification of anamnesis proper to the syllabus of the concerned discipline.
As one devoted to Oriental Studies, the polymathy of the author negotiates the hermeneutic quandary by the surprising inclusion of Lalita Sahasranama. Salutary this is; for mathema as a system of number in Samskrit is at once existence and existent. A certain anonymous vigilance leads the initiate into a tan inverse relation of two adjacencies. Salute we may the restorative energy of this vigilance which approves of our withdrawal from the corybantic refuge from the naive arithmetic world supposedly! Not that I present, nor rather compliment, a de facto eulogy of what we have been sleeping over. The moot question hovers: Is the system as such a commitment in the uncommittedness of what has long been in wait to get acknowledged?
Ramamurthy's vade mecum, yes, I would call it so, on Vedic et. al. numina move from the theoretical vantage point: a harboring stratum of core parody. No longer could it be: 'Calculus, thy name is Newton if not Leibnitz'; for the rudiments in the oversight of occidental inculcation, had never been steered through.
I understand, as would perhaps Spengler or Schrodinger do, that Ramamurthy's effort has, for once, disclosed the infiltration of a kinetic imperative into a bygone facet of impermeable existence, almost like an un-diapered premise of representation entrenching appropriation by Western historicism.
Reading this primer shall tend to generate humility in readership without an accompanying pride of mollification or closure. I wonder and exclaim from some safe distance: "and after so many centuries, we don't know much about this subject". Though it sounds anticlimactic, prefer I do, to put in black and white, as if from a third world, that all advancedness are an accumulation, sadly incompatible with the laden intimations aesthetic, philosophical and religious already there, only to be reminisced by a chosen few of whom Mr. Ramamurthy is for sure, the one.
I commend this Primer Text as one of supreme importance to be a reference amidst an enmeshing dormancy our syllabus is seldom free from and am glad to meditate on its admission to the Jewish Congress Library Catalog by virtue of its exuberance.
"The head sublime, the heart pathos, the genitals beauty, the hands and feet proportion.
As the air to a bird or the sea to a fish,
So is contempt to the contemptible.
The crow wished everything was black,
the owl that everything was white.
Exuberance is Beauty"...
It is beyond doubt that Mathematics, what we have today, owes a huge debt to the outstanding contributions made by Indian Mathematicians over many hundreds, probably thousands, of years. What is quite surprising is that there has been a reluctance to recognise this and one has to conclude that many famous historians of Mathematics found, what they expected to find, or perhaps even what they hoped to find, rather than to realise what was so clear in front of them. The Indian Mathematics was not found just by calculations; Indian Mathematicians could codify it through penance and deep wisdom. The latest evidence in this regard is Vedic Mathematics of 20th Century C.E., obtained through penance by HH Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaj, We can boldly and loudly claim that all concepts of modern day Mathematics have analogues in Indian Mathematics.
The "huge debt" is the beautiful number system invented by the Indians on which most of the Mathematical developments have rested. Laplace puts this as:
"The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way; it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated, when one considers that it was beyond the two greatest men of antiquity - Archimedes and Apollonius".
Histories of Indian Mathematics used to begin by describing the Geometry contained in the Sulbasootras but research into the history of Indian Mathematics has shown that the essentials of this Geometry were older being contained in the altar constructions described in the Vedic mythology text - Shatapatha Brahmana and the Taittreeya Samhita. Also it has been shown that the study of Mathematical Astronomy in India goes back to at least the third millennium B.C.E. and Mathematics and Geometry must have existed to support this study in these ancient times.
The first Mathematics was developed in the Indus valley. The earliest known urban Indian culture was first identified at Harappa in the Punjab and then at Mohenjo-Daro, near the Indus River in the Sindh. Both these sites are now in Pakistan but this is still covered by the term "Indian Mathematics" which, refers to Mathematics developed in the Indian sub-continent. The Indus civilisation (or Harappa civilisation as it is sometimes called) was based in these two cities and also in over a hundred small towns and villages around. It was a civilisation, which began around 2500 B.C.E., and survived until 1700 B.C.E., or even later. The people were literate and used a written script containing around 500 characters, which some have claimed to have deciphered but, being far from claim that this is the case, much research remains to be done before a full appreciation of the Mathematical achievements of this ancient civilisation can be fully assessed.
It is so sad that the due credit has not been provided to Indian Mathematicians and to add fuel to the same, some of the IP or copyrights or fame has been provided to later people who gave the Mathematical concepts, which are already available in Indian Mathematics. One important reason is that most of these texts of Indian Mathematics were given in cryptic form and to decipher the same the students need a deep knowledge of Samskrit and still further knowledge in Mathematics.
At the outset when the author got some feedback on this book as it talks on various things at a very high level like tit-bits. No single volume can be written comprehensively covering entire Indian Mathematics. The very aim of this book is to provide a high level idea of Indian Mathematics and to demystify some of the Mathematical representations and explain the same in simple terms. In this process an attempt has been made to give an overall history of Indian Mathematics also. Again an attempt has been made to, at least, list down the Luminaries of Indian Mathematics in an annexure.
Any discussion involving Mathematics and Samskrit is incomplete without mentioning about Vedic Mathematics. Hence an introductory chapter on Vedic Mathematics has also been included in this book.
Conventions used in this book: The transliterated Samskrit words are written in italics - for instance Bhoota. When Samskrit words are transliterated in English diacritical marks are normally used. However the same has not been used in its entirety in this book, since it makes the reading a little more difficult and since this book is intended for common audience.
Many many thanks from the bottom of my heart are due to Prof. S.A.Sankaranarayanan, who has written a nice introduction and some pleasantries about the author and this edition. I will be failing in my duty if I do not thank Shree. M. Easwaran, who has printed and published this book. He not only does the publication tasks, but also reviews the book and gives vital feedback so that the book is in the fashion it is at present. Even amidst all constraints like soaring power cut, etc., nice job done - thanks a lot. Sincere thanks are due to all those who supported in this noble cause.
Dedication | 5 | |
Introduction | 6 | |
1 | History of Mathmatics in India | 10 |
1.1 | Mathmatics and India | 13 |
1.2 | Numbers and Veda | 17 |
1.3 | Mathmatical Activity in Vedic Period | 24 |
1.4 | Panini and Formal Scientific Notation | 25 |
1.5 | Philosophy and Mathmatics | 26 |
1.6 | The Indian Numeral System | 28 |
1.7 | Siddhantic and Classical Age | 31 |
1.8 | The Decimal System in Harappa | 32 |
1.9 | The Spread of Indian Mathmatics | 32 |
1.10 | The Kerala School | 33 |
1.11 | The sign for Zero | 36 |
1.12 | Zero and the Place Value System | 41 |
1.13 | India and the Scientific Revolution | 45 |
1.14 | Emergence of Calculus | 47 |
1.15 | Importance of Astronomy | 47 |
1.16 | Applied Mathmatics , Solutions to Practical Problems | 50 |
2 | Classical Indian Mathmaticians | 53 |
2.1 | Oral tradition | 61 |
2.2 | Styles of Memorisation | 62 |
2.3 | The Sutra Genre | 63 |
3 | Representation of Numerals in Samskrit | 65 |
3.1 | Katapayadi Sankhya: | 66 |
3.2 | Bhuta Sankhya (Cryptic Method) | 70 |
3.3 | Aryabhatiya Sankhya | 79 |
4 | Applications of Sankhyas | 82 |
4.1 | Carnatic Music: | 83 |
4.2 | Lalita Sahasranama | 86 |
4.3 | Purana-s | 88 |
4.4 | Bala Mantra | 89 |
4.5 | Nama Shastra (naming conventions) | 91 |
5 | Vedic Mathmatics | 94 |
5.1 | Salient Fratures of Vedic Mathmatics | 97 |
5.2 | Formulae and Corollaries of Vedic Mathmatics | 99 |
Annexure 1 | 104 | |
Annexure 2 | 114 | |
Annexure 3 | 117 | |
Bibiliography | 122 |
Dr. Ramamurthy, has been in the forefront of imparting knowledge based studies in the areas of Vedas, Samskrit, Banking, related IT solutions, Information Security, IT Audit and so on. His thirst for continuous learning does not subside. He did research on an unique topic “Information Technology and Samskrit” and obtained Ph.D. – doctorate degree from University of Madras even at the age of late fifties. He has so far penned 18 books on religious literature, Software Testing and Banking topics. Currently he is working on a project of developing a Samskrit based compiler.
Foreword
To text much 'mathema' in a little, multum in parvo, sleep is implicated: by derivation, jagrat is adverted. With a primer of an exclusive kind to which this work belongs, from the post-modern age of hypnoglyph, may the innocents in many of us wake up into a realm of a larger Time vectored 'Vedic', with modalities of dormancy impounded in it by the superstructures of the Western, compliant or not!
If all knowledge be acknowledgement, as Plato would have it, it is good to be ekphrastic like Mr. N. Ramamurthy, by indirection, often tabling that number is neither revelry nor devilry, or geometry is none of the debauchery or carousal of Space. Mere mention of Satapatabrahmana and such and such is gnomic to suggest that when our sleep is profound, we stand and are put in hold! Hu'pnos hedraios, orthoi, nustagmo's
[Epidemics 6.4.15; Littre':310].
My purport is to declaim, though it might seem a bit bizarre, by an analogy: to use the surmise of Galen (129-199 C.E), the late Classical successor of Hippocrates, and the so called advanced most wakeful number-theoreticians of our day may achieve a soothing slumber if they remain erect and vigilant until they begin to nod and wink!
A saying goes: Even Homer nods! If we return to origins, in peripety, we experience a jolt or an un-hinged-ness of our advanced-ness: a constraint to sleepwalk, awakening only when stumble on a stone, paleolithic! I mean, tit-bits are misnomers for even if they pass for so so, cover they do an aura of a decisive juncture of an essential moment, the author attempts to recreate, tending to be minimally narrational. Of course, Laplace supports the choice of Mr. Ramamurthy's locution 'pragnantesten', most pregnant, for a total emergence of mathema's progeny from parent Vedic, with a historic-Indic sense and demystification of not just instants but incidents, relating ensembles of Time and Space. The tit-bit type of primering the profound is endemic to Mr. Ramamurthy's, for his is spiring fulfilled continuity of Shree Chakra resolution.
The contents of this book are a phalanx in an under-coding for a spirited adoption of a subtle differential from our era's norms. Come what may, if a lay starter is schooled in these hundred and odd pages of ages of labor, the durable circadian boundaries blur. Mathematical logic and mathematical rhetoric as more than mere approval of a diaristic discourse must help the learner thank the impersonal chronicle of the theological track of a believer's subjectivity, engendering no muddle in the middle we owe to!
Ramamurthy in his more-than-a-text-book seems to postulate a categorical neither-here-nor-there-ness of what the student world of mathematics has been in. Any way the book is not to be viewed as a recordation calendrically, though there is an overtone of temporal character in the signification of anamnesis proper to the syllabus of the concerned discipline.
As one devoted to Oriental Studies, the polymathy of the author negotiates the hermeneutic quandary by the surprising inclusion of Lalita Sahasranama. Salutary this is; for mathema as a system of number in Samskrit is at once existence and existent. A certain anonymous vigilance leads the initiate into a tan inverse relation of two adjacencies. Salute we may the restorative energy of this vigilance which approves of our withdrawal from the corybantic refuge from the naive arithmetic world supposedly! Not that I present, nor rather compliment, a de facto eulogy of what we have been sleeping over. The moot question hovers: Is the system as such a commitment in the uncommittedness of what has long been in wait to get acknowledged?
Ramamurthy's vade mecum, yes, I would call it so, on Vedic et. al. numina move from the theoretical vantage point: a harboring stratum of core parody. No longer could it be: 'Calculus, thy name is Newton if not Leibnitz'; for the rudiments in the oversight of occidental inculcation, had never been steered through.
I understand, as would perhaps Spengler or Schrodinger do, that Ramamurthy's effort has, for once, disclosed the infiltration of a kinetic imperative into a bygone facet of impermeable existence, almost like an un-diapered premise of representation entrenching appropriation by Western historicism.
Reading this primer shall tend to generate humility in readership without an accompanying pride of mollification or closure. I wonder and exclaim from some safe distance: "and after so many centuries, we don't know much about this subject". Though it sounds anticlimactic, prefer I do, to put in black and white, as if from a third world, that all advancedness are an accumulation, sadly incompatible with the laden intimations aesthetic, philosophical and religious already there, only to be reminisced by a chosen few of whom Mr. Ramamurthy is for sure, the one.
I commend this Primer Text as one of supreme importance to be a reference amidst an enmeshing dormancy our syllabus is seldom free from and am glad to meditate on its admission to the Jewish Congress Library Catalog by virtue of its exuberance.
"The head sublime, the heart pathos, the genitals beauty, the hands and feet proportion.
As the air to a bird or the sea to a fish,
So is contempt to the contemptible.
The crow wished everything was black,
the owl that everything was white.
Exuberance is Beauty"...
It is beyond doubt that Mathematics, what we have today, owes a huge debt to the outstanding contributions made by Indian Mathematicians over many hundreds, probably thousands, of years. What is quite surprising is that there has been a reluctance to recognise this and one has to conclude that many famous historians of Mathematics found, what they expected to find, or perhaps even what they hoped to find, rather than to realise what was so clear in front of them. The Indian Mathematics was not found just by calculations; Indian Mathematicians could codify it through penance and deep wisdom. The latest evidence in this regard is Vedic Mathematics of 20th Century C.E., obtained through penance by HH Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaj, We can boldly and loudly claim that all concepts of modern day Mathematics have analogues in Indian Mathematics.
The "huge debt" is the beautiful number system invented by the Indians on which most of the Mathematical developments have rested. Laplace puts this as:
"The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way; it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated, when one considers that it was beyond the two greatest men of antiquity - Archimedes and Apollonius".
Histories of Indian Mathematics used to begin by describing the Geometry contained in the Sulbasootras but research into the history of Indian Mathematics has shown that the essentials of this Geometry were older being contained in the altar constructions described in the Vedic mythology text - Shatapatha Brahmana and the Taittreeya Samhita. Also it has been shown that the study of Mathematical Astronomy in India goes back to at least the third millennium B.C.E. and Mathematics and Geometry must have existed to support this study in these ancient times.
The first Mathematics was developed in the Indus valley. The earliest known urban Indian culture was first identified at Harappa in the Punjab and then at Mohenjo-Daro, near the Indus River in the Sindh. Both these sites are now in Pakistan but this is still covered by the term "Indian Mathematics" which, refers to Mathematics developed in the Indian sub-continent. The Indus civilisation (or Harappa civilisation as it is sometimes called) was based in these two cities and also in over a hundred small towns and villages around. It was a civilisation, which began around 2500 B.C.E., and survived until 1700 B.C.E., or even later. The people were literate and used a written script containing around 500 characters, which some have claimed to have deciphered but, being far from claim that this is the case, much research remains to be done before a full appreciation of the Mathematical achievements of this ancient civilisation can be fully assessed.
It is so sad that the due credit has not been provided to Indian Mathematicians and to add fuel to the same, some of the IP or copyrights or fame has been provided to later people who gave the Mathematical concepts, which are already available in Indian Mathematics. One important reason is that most of these texts of Indian Mathematics were given in cryptic form and to decipher the same the students need a deep knowledge of Samskrit and still further knowledge in Mathematics.
At the outset when the author got some feedback on this book as it talks on various things at a very high level like tit-bits. No single volume can be written comprehensively covering entire Indian Mathematics. The very aim of this book is to provide a high level idea of Indian Mathematics and to demystify some of the Mathematical representations and explain the same in simple terms. In this process an attempt has been made to give an overall history of Indian Mathematics also. Again an attempt has been made to, at least, list down the Luminaries of Indian Mathematics in an annexure.
Any discussion involving Mathematics and Samskrit is incomplete without mentioning about Vedic Mathematics. Hence an introductory chapter on Vedic Mathematics has also been included in this book.
Conventions used in this book: The transliterated Samskrit words are written in italics - for instance Bhoota. When Samskrit words are transliterated in English diacritical marks are normally used. However the same has not been used in its entirety in this book, since it makes the reading a little more difficult and since this book is intended for common audience.
Many many thanks from the bottom of my heart are due to Prof. S.A.Sankaranarayanan, who has written a nice introduction and some pleasantries about the author and this edition. I will be failing in my duty if I do not thank Shree. M. Easwaran, who has printed and published this book. He not only does the publication tasks, but also reviews the book and gives vital feedback so that the book is in the fashion it is at present. Even amidst all constraints like soaring power cut, etc., nice job done - thanks a lot. Sincere thanks are due to all those who supported in this noble cause.
Dedication | 5 | |
Introduction | 6 | |
1 | History of Mathmatics in India | 10 |
1.1 | Mathmatics and India | 13 |
1.2 | Numbers and Veda | 17 |
1.3 | Mathmatical Activity in Vedic Period | 24 |
1.4 | Panini and Formal Scientific Notation | 25 |
1.5 | Philosophy and Mathmatics | 26 |
1.6 | The Indian Numeral System | 28 |
1.7 | Siddhantic and Classical Age | 31 |
1.8 | The Decimal System in Harappa | 32 |
1.9 | The Spread of Indian Mathmatics | 32 |
1.10 | The Kerala School | 33 |
1.11 | The sign for Zero | 36 |
1.12 | Zero and the Place Value System | 41 |
1.13 | India and the Scientific Revolution | 45 |
1.14 | Emergence of Calculus | 47 |
1.15 | Importance of Astronomy | 47 |
1.16 | Applied Mathmatics , Solutions to Practical Problems | 50 |
2 | Classical Indian Mathmaticians | 53 |
2.1 | Oral tradition | 61 |
2.2 | Styles of Memorisation | 62 |
2.3 | The Sutra Genre | 63 |
3 | Representation of Numerals in Samskrit | 65 |
3.1 | Katapayadi Sankhya: | 66 |
3.2 | Bhuta Sankhya (Cryptic Method) | 70 |
3.3 | Aryabhatiya Sankhya | 79 |
4 | Applications of Sankhyas | 82 |
4.1 | Carnatic Music: | 83 |
4.2 | Lalita Sahasranama | 86 |
4.3 | Purana-s | 88 |
4.4 | Bala Mantra | 89 |
4.5 | Nama Shastra (naming conventions) | 91 |
5 | Vedic Mathmatics | 94 |
5.1 | Salient Fratures of Vedic Mathmatics | 97 |
5.2 | Formulae and Corollaries of Vedic Mathmatics | 99 |
Annexure 1 | 104 | |
Annexure 2 | 114 | |
Annexure 3 | 117 | |
Bibiliography | 122 |