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Books > History > Ancient > Ancient Indian Mathematics (With Special Reference to Vedic Mathematics and Astronomy)

Ancient Indian Mathematics (With Special Reference to Vedic Mathematics and Astronomy)

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**Foreword**

The ancient Indian mathematicians enriched the subject over the ages, by their outstanding contributions. From the times of the Indus Valley civilization (3500 B.C.E.) up to the times of Islamic conquest, Indian scientists and mathematicians were leaders in many different areas of knowledge. They stood singularly apart in mathematics. India takes a comfortable and secure place of legitimate pride with other ancient civilizations of the world: Egypt, China, Mesopotamia, and Greece.

It is surprising that in our universities and colleges students, and teachers too in some cases, are not aware of the many notable achievements of our ancestors in the field of science in general and of mathematics in particular. The historians of science should share the blame as well, because of their reluctance to properly assess the quantum of contributions to science made by the ancient scholars of India. Perhaps, we are still haunted by the ghost of T.B. Macaulay (1800-1859), who designed a pattern of education for the British India to train “a class of persons, Indian in blood and colour but English in taste, in opinion, in morals, and in intellect”.

Many a time, injustice has been meted out to and harm is perpetrated against the wisdom of our ancient scientists. We can cite a number of instances; one would be enough for the present. Take for example the decimal system of numerals originated in India. In fact, when Arab scholars came to know about this system of numerals, they hailed them as the numbers from India. The Father of Algebra, al Khwarizme, in 825 C.E., wrote an essay on The Calculation with Hindu Numerals. Al Khwarizmi was the coordinating link between India and Arabic mathematics one of those “irreplaceable men capable of facing in two directions at once”, Abul Hasan Ahmed ibn Ibrahim al Uqlidisi (920-980 C.E.), an Arab mathematician, whose Kitab al fusul fi ah-hisab al Hindi (Chapters in Indian Mathematics) is an important known Arabic work discussing the positional use of decimal system of numerals that the Arabs got from India. Giving credit to India for inventing and to the Arabs for popularizing, historians started calling these numerals as Hindu-Arabic Numerals.

In spite of the fact that the world acknowledges and recognizes the origins of the decimal system, there are scholars who still call these numbers as Arabic numerals and a few in recent times went to the extent of labeling them as “eurobic'' numerals. The world knows that during the 12th Century C.E., an enterprising Italian merchant and scholar, Fibonacci, took these numerals to Italy from Arab countries and popularized them in the whole of Europe. Therefore, should they be called “eurobic” numerals? It should be recalled what Fibonacci wrote “compared to the method of Indians, all other methods is a mistake.” This method of Indians is none other than our very simple arithmetic of addition, subtraction, multiplication, and division.

Famous scientists and men of letters hailed these decimal numerals as the greatest contribution by India to the onward march of human civilization. Laplace (1749-1827 C.E.), France, wrote: “It is India that gave us the ingenious method of expressing all numbers by the means of ten symbols, each symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.

Irfah (1992), France, said, “the measure of genius ofIndian civilization to which we owe our modern [number] system, is all the greater in that it was the only one in all history to have achieved this triumph.” He (1994) also wrote, “A thousand years ahead of Europeans, Indian Savants knew that zero and infinity were mutually inverse notions”.

Bourbaki (1998), France, wrote, “modern mathematics was known during the medieval times as Modus Indorium or method of the Indians.” Bourbaki, further added, “our decimal system which [by the agency of the Arabs] is derived from Hindu mathematics, where its use is attested already from the first century to our era.”

Famous historian of mathematics, F. Cajori wrote that he and others “suspect that Diophantus got his first glimpse of algebraic knowledge from India.”

Albert Einstein (1879-1955 C.E.), U.S.A., said, “we owe a lot to Indians who taught us how to count, without which no worthwhile scientific discovery could have been made.”

A.L. Basham (20th Century), Australia, said, “ ... the world owes most to India in the realm of mathematics ... which was developed in the Gupta period, to a shape more advanced than that reached by other nations of antiquity.”

Voltaire (18th Century), France, speaking on Indian mathematics and spiritualism said, “I am convinced that everything has come down to us from the banks of Ganga; Astronomy, Astrology, and Spiritualism. It is very important to note that 2500 years ago, at the least, Pythagoras went from Samos to the Ganga to learn Geometry.”

T. Dantzig (1884 - 1956), U.S.A, wrote, “ the invention of zero will always stand out as one of the greatest simple achievements of the human race.” He further added, “Long period of nearly five thousand years ... the history of reckoning presents a peculiar picture of desolate stagnation. When viewed in this light, the achievements of the unknown Hindu, who some time in the first centuries of our era discovered the principle of position, assumes the importance of a world event.”

The Indian p lace-system of numerals might have spread first to the nearby Persia and from there to the Arab land. In 662 CE, a Nestorian bishop living in what is now called Iraq said, “I will omit all discussion of the science of the Indians ... of their subtle discoveries in astronomy - discoveries that are more ingenious than those of the Greeks and the Babylonians - and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe that because they speak Greek they have arrived at the limits of science would read the Indian texts, they would be convinced even if a little late in the day that there are others who know something of value.”

G. Halstead said, “the importance of the creation of zero mark can never be exaggerated, giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power is the characteristic power of the Hindu race from whence it sprang. No single mathematical creation has been more potent for the general on go of intelligence and power.”

Indian mathematics has its roots in Vedic literature, which is about 4,000 years old. For example, in the anuvaka section of Taittiriya Upanishad of Yajurveda, we find a precise sequence of multiplication of numbers by 100. It shows without doubt that the Vedic scholars were comfortable with the usage of very large numbers.

The earliest concept of a heliocentric model of the solar system, in which the sun is at the centre of the system with the earth orbiting the sun , is found in the Vedic text. The Aithareya Brahmana ( 8th Century BCE) states, “The sun never sets nor rises. When people think the sun is setting, it is not so, they are mistaken.”

The oldest mathematical manuscript written in Buddhist hybrid Sanskrit timed as belonging to early centuries of the common era, was discovered in 1881 in the village of Bakshali, now in Pakistan. The Bakshali manuscript also “employs a decimal place value system with a dot for zero.”

**Contents**

Vice Chancellor s Message | ||

Foreword | i | |

About the Chief Editor | x | |

About the Editors - Articles | xi | |

1. RE-ENVISIONING THE DYNAMISM OF GEOMETRICAL PERSUITS OF INDIAN ANTIQUITY - Dilip Kumar Sinha | 1 | |

2. PAVULURJ MALLANA - OUTLINES OF HIS LIFE AND WORK P.V. Arunachalam | 8 | |

3 | A GREEDY ALGORITHM HIDDEN IN SULVASUTRA- Kannan and Sharan Gopal | 17 |

4 | ZERO- Gokulanand Das | 27 |

5 | IS THERE MATHEMATICS IN THE VEDAS ?-K.V. Krishna Murty | 35 |

6 | REVISITING INDIAN MATHEMATICAL LEGACY WITH ITS EVOLVING NEXUS WITH TIMES - Dilip Kumar Sinha | 41 |

7 | A GENERALISED APPROACH FOR FINDING THE Nth ORDER ROOTS OF NUMBERS - R.V.S.S. Avadhanulu | 45 |

8 | EXCERPTS FROM SOME MODERN TEXTS ON ANCIENT INDIAN MATHEMATICS - S. Bhargava | 51 |

9 | DE-CODING CRYPTOGRAPHICARYABHATIYA VALUES FOR NUMBER OF REVOLUTIONS OF GEO-CENTRIC PLANETS IN A MAHAYUGA (43,20,000 YEARS) AND COMPARISON OF THEIR SIDEREAL PERIODS WITH THEIR PRESENT DAY VALUES - - Venkatesha Murthy | 66 |

10 | VERIFICATION OF CAKRAVALA METHOD OF JAYADEVA - BHASKARA II FOR SOLVING NX2 + 1 = Y2THROUGH MODIFIED CONTINUED FRACTION EXPANSION METHOD - Venkatesha Murthy | 78 |

11 | SANKARA'S GEOMETRICAL APPROACH TO CITRABHANU'S EKAVIMSATI PRASNOTTARA V. Madhukar Mallayya | 99 |

12 | DEVELOPMENT OF COMBINATORICS FROM THE PRATYAYAS IN SANSKRIT PROSODY - Venugopal.D. Heroor | 128 |

13 | IMPROVED SIDDHANTIC PROCEDURE FOR LUNAR AND SOLAR ECLIPSES. - S.Balachandra Rao. | 169 |

14 | TRANSITS OF VENUS AND MERCURY AND OCCULTATION Padmaja Venugopal | 186 |

15 | JYA: CONCEPT, PROPERTIES AND APPLICATIONS - S. Madhavan | 205 |

16 | THE SINE FUNCTION IN ANCIENT INDIAN MATHEMATICS - Shailesh A Shirali | 228 |

Item Code:

NAH304

Cover:

Hardcover

Edition:

2011

Publisher:

Language:

Sanskrit Text with English Translation

Size:

10 inch x 7 inch

Pages:

262

Other Details:

Weight of the Book: 680 gms

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**Foreword**

The ancient Indian mathematicians enriched the subject over the ages, by their outstanding contributions. From the times of the Indus Valley civilization (3500 B.C.E.) up to the times of Islamic conquest, Indian scientists and mathematicians were leaders in many different areas of knowledge. They stood singularly apart in mathematics. India takes a comfortable and secure place of legitimate pride with other ancient civilizations of the world: Egypt, China, Mesopotamia, and Greece.

It is surprising that in our universities and colleges students, and teachers too in some cases, are not aware of the many notable achievements of our ancestors in the field of science in general and of mathematics in particular. The historians of science should share the blame as well, because of their reluctance to properly assess the quantum of contributions to science made by the ancient scholars of India. Perhaps, we are still haunted by the ghost of T.B. Macaulay (1800-1859), who designed a pattern of education for the British India to train “a class of persons, Indian in blood and colour but English in taste, in opinion, in morals, and in intellect”.

Many a time, injustice has been meted out to and harm is perpetrated against the wisdom of our ancient scientists. We can cite a number of instances; one would be enough for the present. Take for example the decimal system of numerals originated in India. In fact, when Arab scholars came to know about this system of numerals, they hailed them as the numbers from India. The Father of Algebra, al Khwarizme, in 825 C.E., wrote an essay on The Calculation with Hindu Numerals. Al Khwarizmi was the coordinating link between India and Arabic mathematics one of those “irreplaceable men capable of facing in two directions at once”, Abul Hasan Ahmed ibn Ibrahim al Uqlidisi (920-980 C.E.), an Arab mathematician, whose Kitab al fusul fi ah-hisab al Hindi (Chapters in Indian Mathematics) is an important known Arabic work discussing the positional use of decimal system of numerals that the Arabs got from India. Giving credit to India for inventing and to the Arabs for popularizing, historians started calling these numerals as Hindu-Arabic Numerals.

In spite of the fact that the world acknowledges and recognizes the origins of the decimal system, there are scholars who still call these numbers as Arabic numerals and a few in recent times went to the extent of labeling them as “eurobic'' numerals. The world knows that during the 12th Century C.E., an enterprising Italian merchant and scholar, Fibonacci, took these numerals to Italy from Arab countries and popularized them in the whole of Europe. Therefore, should they be called “eurobic” numerals? It should be recalled what Fibonacci wrote “compared to the method of Indians, all other methods is a mistake.” This method of Indians is none other than our very simple arithmetic of addition, subtraction, multiplication, and division.

Famous scientists and men of letters hailed these decimal numerals as the greatest contribution by India to the onward march of human civilization. Laplace (1749-1827 C.E.), France, wrote: “It is India that gave us the ingenious method of expressing all numbers by the means of ten symbols, each symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.

Irfah (1992), France, said, “the measure of genius ofIndian civilization to which we owe our modern [number] system, is all the greater in that it was the only one in all history to have achieved this triumph.” He (1994) also wrote, “A thousand years ahead of Europeans, Indian Savants knew that zero and infinity were mutually inverse notions”.

Bourbaki (1998), France, wrote, “modern mathematics was known during the medieval times as Modus Indorium or method of the Indians.” Bourbaki, further added, “our decimal system which [by the agency of the Arabs] is derived from Hindu mathematics, where its use is attested already from the first century to our era.”

Famous historian of mathematics, F. Cajori wrote that he and others “suspect that Diophantus got his first glimpse of algebraic knowledge from India.”

Albert Einstein (1879-1955 C.E.), U.S.A., said, “we owe a lot to Indians who taught us how to count, without which no worthwhile scientific discovery could have been made.”

A.L. Basham (20th Century), Australia, said, “ ... the world owes most to India in the realm of mathematics ... which was developed in the Gupta period, to a shape more advanced than that reached by other nations of antiquity.”

Voltaire (18th Century), France, speaking on Indian mathematics and spiritualism said, “I am convinced that everything has come down to us from the banks of Ganga; Astronomy, Astrology, and Spiritualism. It is very important to note that 2500 years ago, at the least, Pythagoras went from Samos to the Ganga to learn Geometry.”

T. Dantzig (1884 - 1956), U.S.A, wrote, “ the invention of zero will always stand out as one of the greatest simple achievements of the human race.” He further added, “Long period of nearly five thousand years ... the history of reckoning presents a peculiar picture of desolate stagnation. When viewed in this light, the achievements of the unknown Hindu, who some time in the first centuries of our era discovered the principle of position, assumes the importance of a world event.”

The Indian p lace-system of numerals might have spread first to the nearby Persia and from there to the Arab land. In 662 CE, a Nestorian bishop living in what is now called Iraq said, “I will omit all discussion of the science of the Indians ... of their subtle discoveries in astronomy - discoveries that are more ingenious than those of the Greeks and the Babylonians - and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe that because they speak Greek they have arrived at the limits of science would read the Indian texts, they would be convinced even if a little late in the day that there are others who know something of value.”

G. Halstead said, “the importance of the creation of zero mark can never be exaggerated, giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power is the characteristic power of the Hindu race from whence it sprang. No single mathematical creation has been more potent for the general on go of intelligence and power.”

Indian mathematics has its roots in Vedic literature, which is about 4,000 years old. For example, in the anuvaka section of Taittiriya Upanishad of Yajurveda, we find a precise sequence of multiplication of numbers by 100. It shows without doubt that the Vedic scholars were comfortable with the usage of very large numbers.

The earliest concept of a heliocentric model of the solar system, in which the sun is at the centre of the system with the earth orbiting the sun , is found in the Vedic text. The Aithareya Brahmana ( 8th Century BCE) states, “The sun never sets nor rises. When people think the sun is setting, it is not so, they are mistaken.”

The oldest mathematical manuscript written in Buddhist hybrid Sanskrit timed as belonging to early centuries of the common era, was discovered in 1881 in the village of Bakshali, now in Pakistan. The Bakshali manuscript also “employs a decimal place value system with a dot for zero.”

**Contents**

Vice Chancellor s Message | ||

Foreword | i | |

About the Chief Editor | x | |

About the Editors - Articles | xi | |

1. RE-ENVISIONING THE DYNAMISM OF GEOMETRICAL PERSUITS OF INDIAN ANTIQUITY - Dilip Kumar Sinha | 1 | |

2. PAVULURJ MALLANA - OUTLINES OF HIS LIFE AND WORK P.V. Arunachalam | 8 | |

3 | A GREEDY ALGORITHM HIDDEN IN SULVASUTRA- Kannan and Sharan Gopal | 17 |

4 | ZERO- Gokulanand Das | 27 |

5 | IS THERE MATHEMATICS IN THE VEDAS ?-K.V. Krishna Murty | 35 |

6 | REVISITING INDIAN MATHEMATICAL LEGACY WITH ITS EVOLVING NEXUS WITH TIMES - Dilip Kumar Sinha | 41 |

7 | A GENERALISED APPROACH FOR FINDING THE Nth ORDER ROOTS OF NUMBERS - R.V.S.S. Avadhanulu | 45 |

8 | EXCERPTS FROM SOME MODERN TEXTS ON ANCIENT INDIAN MATHEMATICS - S. Bhargava | 51 |

9 | DE-CODING CRYPTOGRAPHICARYABHATIYA VALUES FOR NUMBER OF REVOLUTIONS OF GEO-CENTRIC PLANETS IN A MAHAYUGA (43,20,000 YEARS) AND COMPARISON OF THEIR SIDEREAL PERIODS WITH THEIR PRESENT DAY VALUES - - Venkatesha Murthy | 66 |

10 | VERIFICATION OF CAKRAVALA METHOD OF JAYADEVA - BHASKARA II FOR SOLVING NX2 + 1 = Y2THROUGH MODIFIED CONTINUED FRACTION EXPANSION METHOD - Venkatesha Murthy | 78 |

11 | SANKARA'S GEOMETRICAL APPROACH TO CITRABHANU'S EKAVIMSATI PRASNOTTARA V. Madhukar Mallayya | 99 |

12 | DEVELOPMENT OF COMBINATORICS FROM THE PRATYAYAS IN SANSKRIT PROSODY - Venugopal.D. Heroor | 128 |

13 | IMPROVED SIDDHANTIC PROCEDURE FOR LUNAR AND SOLAR ECLIPSES. - S.Balachandra Rao. | 169 |

14 | TRANSITS OF VENUS AND MERCURY AND OCCULTATION Padmaja Venugopal | 186 |

15 | JYA: CONCEPT, PROPERTIES AND APPLICATIONS - S. Madhavan | 205 |

16 | THE SINE FUNCTION IN ANCIENT INDIAN MATHEMATICS - Shailesh A Shirali | 228 |

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