– Vedanga Jyotisa.

In mental abstraction and concentration of thought the Hindus are proverbially happy. Apart from direct testimony on the point, the literature of the Hindus furnishes unmistakable evidence to prove that the ancient Hindus possessed astonishing power of memory and concentration of thought. The science of mathematics, the most abstract of all sciences, must have an irresistible fascination for the minds of the Hindus.

The great German critic, Schlegel wrote in his History of Literature, p. 123: "The decimal cyphers, the honor of which, next to letters the most important of human discoveries, has, with the common consent of historical authorities, been ascribed to Hindus."

Mathematics is the science to which Indians have contributed the most. Our decimal system, place notation, numbers 1 through 9, and the ubiquitous 0, are all major Indian contributions to world science. Without them, our modern world of computer sciences, earth-launched satellites, microchips, and artificial intelligence would all have been impossible.

(source: An Introduction to India - By Stanley Wolpert p. 194).

Hermann Hankel (1839 - 1873) born in Halle, Germany in his History of Mathematics says:

“ It is remarkable to what extent Indian Mathematics enters into the Science of our time”

(source: Is India Civilized? - Essays on Indian Culture - By Sir John Woodroffe Ganesh & Co. Publishers 1922 p. 182).

The earliest recorded Indian mathematics was found along the banks of the Indus. Archaeologists have uncovered several scales, instruments, and other measuring devices. The Harappans employed a variety of plumb bobs that reveal a system of weights 27.584 grams. If we assign that a value of 1, other weights scale in at .05, .1, .2, .5, 2, 5, 10, 20, 50, 100, 200 and 500. These weights have been found in sites that span a five-thousand-year period, with little change in size.

Archaeologists also found a “ruler” made of shell lines drawn 6.7 millimeters apart with a high degree of accuracy. Two of the lines are distinguished by circles and are separated by 33.5 millimeters, or 1.32 inches. This distance is the so-called Indus inch.

(source: Lost Discoveries - Dick Teresi p. 59).

Dr. David Gray writes:

"The study of mathematics in the West has long been characterized by a certain ethnocentric bias, a bias which most often manifests not in explicit racism, but in a tendency toward undermining or eliding the real contributions made by non-Western civilizations. The debt owed by the West to other civilizations, and to India in particular, go back to the earliest epoch of the "Western" scientific tradition, the age of the classical Greeks, and continued up until the dawn of the modern era, the renaissance, when Europe was awakening from its dark ages."

Dr Gray goes on to list some of the most important developments in the history of mathematics that took place in India, summarizing the contributions of luminaries such as Aryabhatta, Brahmagupta, Mahavira, Bhaskara and Maadhava. He concludes by asserting that "the role played by India in the development (of the scientific revolution in Europe) is no mere footnote, easily and inconsequentially swept under the rug of Eurocentric bias. To do so is to distort history, and to deny India one of its greatest contributions to world civilization."

Mathematics and Music:

Pingala (3rd C AD), author of Chandasutra explored the relationship between combinatorics and musical theory anticipating Mersenne (1588-1648) author of a classic on musical theory.

His contributions to mathematics include:

The formation of a matrix.
Invention of the binary number system (while he was forming a matrix for musical purposes).
The concept of a binary code, similar to Morse code.
First use of the Fibonacci sequence.
First use of Pascal's triangle, which he refers to as Meru-prastaara.
Used a dot (.) to denote zero.
His work, along with Panini's work, was foundational to the development of computing.

(source: Science and Mathematics in India). Refer to chapter on Hindu Music and Indian Mathematics.

Fascination with numbers has been an abiding characteristic of Indian civilization, not only large numbers but very small ones as well. Operations with zero attracted the interest of both Bhaskaracharya (b. 1114) and Srinivas Ramanujan (1887-1920).

In Ramayana, the great Indian epic, there is a description of two armies facing, each other. The size of the larger army led by Rama is given as follows in a 17th century translation of the epic by Kottayam Kerala varma Thampuran:

Hundred hundred thousands make a Crore
Hundred thousand crores make a Sankhu
Hundred thousand sankhus make a Maha-sankhu
Hundred thousand maha-sankhus make a Vriundam
Hundred thousand vriundam make a Maha-vriundam
Hundred thousand maha-vriundams make a Padmam
Hundred thousand padmams make a Maha-padmam
Hundred thousand maha-padmams make a Kharvam
Hundred thousand kharvams make a Maha-kharvam
Hundred thousand maha-kharvams make a Samudra
Hundred thousand samudras make a Maha-ougham.

The importance of number names in the evolution of the decimal place value notation in India cannot be exaggerated. The word-numeral system was the logical outcome of proceeding by multiples of ten. Thus, in an early system, 60,799 is denoted by the Sanskrit word sastim (60), shsara (thousand), sapta (seven) satani (hundred), navatim (nine ten times) and nava (nine). Such a system presupposes a scientifically based vocabulary of number names in which the principles of addition, subtraction and multiplication are used. It requires:

1. the naming of the first nine digits (eka, dvi, tri, catur, pancha, sat, sapta, asta, nava);
2. a second group of nine numbers obtained by multiplying each of the nine digits in 1 by ten (dasa, vimsat, trimsat, catvarimsat, panchasat, sasti, saptati, astiti, navati): and
3. a group of numbers which are increasing integral powers of 10, starting with 102 (satam sagasara, ayut, niyuta, prayuta, arbuda, nyarbuda, samudra, Madhya, anta, parardha…).

To understand why word numerals persisted in India, even after the Indian numerals became widespread, it is necessary to recognize the importance of the oral mode of preserving and disseminating knowledge. An important characteristic of written texts in India from times immemorial was the sutra style of writing, which presented information in a cryptic form, leaving out details and rationale to be filled in by teachers and commentators. In short pithy sentences, often expressed in verse, the sutras enabled the reader to memorize the content easily.
(source: The crest of the peacock: Non-European roots of Mathematics - By George Gheverghese Joseph p.401 - 403).

In the Vedic age, India was ahead of the rest in mathematics and astronomy. Thus, the geometry of the Shulba Sutras (The Rules of the Cord), geometrical appendices to the manuals of ritual (Shrauta Sutras) include the oldest known formulation of the theorem named after Pythagoras, developed in the context of Vedic altar-building. The first decimal system and the oldest names of "astronomical" numbers such as quadrillions and quintillions. Arabs still call the decimal system rakmu 'l-Hind, from Hind, "India."

(source: Mathematics as Known to the Vedic Samhitas - By M. D. Pandit p. 20).

Highly intellectual and given to abstract thinking as they were, one would expect the ancient Indians to excel in mathematics. Ancient Indians developed a system of mathematics far superior, to that of the Greeks. Ancient Vedic mathematicians devised sutras for solving mathematical problems with apparent ease. Among the most vital parts of our heritage are the numerals and the decimal system. The miscalled "Arabic" numerals are found on the Rock Edicts of Ashoka (250 B.C.), a thousand years before their occurrence in Arabic literature. Hindsaa (numerals) in Arabic means from India. Jawaharlal Nehru has said, " The clumsy method of using a counting frame and the use of Roman and such like numerals had long retarded progress when the ten Indian numerals, including the zero sign, liberated the human mind from these restrictions and threw a flood of light on the behavior of numbers."

(source: The Discovery of India - By Jawaharlal Nehru Oxford University Press. 1995 p. 216).

Vedanga Jyotisa says "As are the crests on the heads of peacocks, as are the gems on the hoods of the snakes so is the ganita (Mathematics) at the top of the sciences known as Vedanga. In this period, ganita is a comprehensive term which included arithmetic, algebra and astronomy. Geometry was also investigated but was placed in a different general science known as kalpa. Indians were the first to use the decimal either to increase or decrease the value of the figure which was presided by Laplace, the great French mathematician. Indians were the first to use the 'zero' as a symbol in mathematics. They invented the present numerical system. India teachers taught arithmetic and algebra, Vedic Sulva Sutras were earlier than the Alexandrian geometry of Hero. The earliest available work was Bakshali Manuscript. Ganita-Sara-Sangraham of Mahavira acarya who lived between Brahmagupta and Bhaskaracharya.

The 'Pythagoras theorem' which stated in Sulva Sutras by Baudhayana's (6th century C. E): "The diagonal of a rectangle produces both areas, which its length and breadth produce separately." Arya Bhatta discovered the method of finding out the areas of a triangle, a trapezium and a circle. The approximate value of an 'irrational number' i.e. 2 (dvikarani) (1.143256) and 3 (1.7320513) can be obtained, Baudhayana and Apastamba.

In the geometry of the circle, "Arybhatta I" gave a value for pi (tyajya) which is correct to the four decimal places in a sloka (Sankara Varman's treatise on astronomy, Sadratnamala) theorems and their deductions:

"Lemma of Brahmagupta for integral solution or the indeterminate equation of second degree. John Pell (1611-1685) discovered this in the 17th century. Indians discovered it a 1,000 years earlier.

(source: Hinduism and Scientific Quest - By T R. R. Iyengar p. 151-152).

The most fundamental contribution of ancient India in mathematics is the invention of decimal system of enumeration, including the invention of zero. The decimal system uses nine digits (1 to 9) and the symbol zero (for nothing) to denote all natural numbers by assigning a place value to the digits. The Arabs carried this system to Africa and Europe. The Vedas and Valmiki Ramayana used this system, though the exact dates of these works are not known. MohanjoDaro and Harappa excavations (which may be around 3000 B.C. old) also give specimens of writing in India. Aryans came 1000 years later, around 2000 B.C. Being very religious people, they were deeply interested in planetary positions to calculate auspicious times, and they developed astronomy and mathematics towards this end. They identified various nakshatras (constellations) and named the months after them. They could count up to 1012, while the Greeks could count up to 104 and Romans up to 108. Values of irrational numbers were also known to them to a high degree of approximation. Pythagoras Theorem can be also traced to the Aryan's Sulbasutras. These Sutras, estimated to be between 800 B.C. and 500 B.C., cover a large number of geometric principles.

Said the great and magnanimous Pierre Simon de Laplace, (1749-1827) French mathematician, philosopher, and astronomer, a contemporary of Napoleon :

" It is India that gave us the ingenious method of expressing all numbers by ten symbols, each 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 the more when we remember that it escaped the genius of Archimedes and Appollnius, two of the greatest men produced by antiquity."

(source: The Discovery of India - By Jawaharlal Nehru Oxford University Press. 1995 p. 217).

Brilliant as it was, this invention was no accident. In the Western world, the cumbersome Roman numeral system posed as a major obstacle, and in China the pictorial script posed as a hindrance. But in India, almost everything was in place to favor such a development. There was already a long and established history in the use of decimal numbers, and philosophical and cosmological constructs encouraged a creative and expansive approach to number theory. Panini's studies in linguistic theory and formal language and the powerful role of symbolism and representational abstraction in art and architecture may have also provided an impetus, as might have the rationalist doctrines and the exacting epistemology of the Nyaya Sutras, and the innovative abstractions of the Syadavada and Buddhist schools of learning.

Panini and Formal Scientific Notation

A particularly important development in the history of Indian science that was to have a profound impact on all mathematical treatises that followed was the pioneering work by Panini (6th C BC) in the field of Sanskrit grammar and linguistics. Besides expounding a comprehensive and scientific theory of phonetics, phonology and morphology, Panini provided formal production rules and definitions describing Sanskrit grammar in his treatise called Asthadhyayi. Basic elements such as vowels and consonants, parts of speech such as nouns and verbs were placed in classes. The construction of compound words and sentences was elaborated through ordered rules operating on underlying structures in a manner similar to formal language theory.

Today, Panini's constructions can also be seen as comparable to modern definitions of a mathematical function. G G Joseph, in The crest of the peacock argues that the algebraic nature of Indian mathematics arises as a consequence of the structure of the Sanskrit language. Ingerman in his paper titled Panini-Backus form finds Panini's notation to be equivalent in its power to that of Backus - inventor of the Backus Naur Form used to describe the syntax of modern computer languages. Thus Panini's work provided an example of a scientific notational model that could have propelled later mathematicians to use abstract notations in characterizing algebraic equations and presenting algebraic theorems and results in a scientific format.

(source: Science and Mathematics in India).

The decimal system was known to Aryabhatta and Brahmagupta long before its appearance in the writings of the Arabs and the Syrians; it was adopted by China from Buddhist missionaries; and Muhammad Ibn Musa al-Khwarazni, the greatest mathematician of his age (ca 850 A.D.), seems to have introduced it into Baghdad.

Zero, this most modest and most valuable of all numerals is one of the subtle gifts of India to mankind. The earliest use of the zero symbol, so far discovered, is in one of the scriptural books dated about 200 B.C. The zero, called shunya or nothing, was originally a dot and later it became a small circle. It was considered as a number like any other. Professor G. B. Halsted, in his book ' Mathematics for the Million' (London 1942) thus emphasizes the vital significance of this invention:

"The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol but helpful power, is the characteristic of the Hindu race whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power." It was India that first domesticated zero, through the Hindu familiarity with the concepts of infinity and the void. Neither pagan Rome nor the Christian Europe of the Middle Ages had any truck with it. It's all, as the Hindus knew, a play between the void and the absolute.

Yet another modern mathematician has grown eloquent over this historic event. Dantzig in his 'Number' writes:

"This long period of nearly five thousand years saw the rise and fall of many a civilization, each leaving behind a heritage of literature, art, philosophy, and religion. But what was the net achievement in the field of reckoning, the earliest art practiced by man? An inflexible numeration so crude as to make progress well nigh impossible, and a calculating device so limited in scope that even elementary calculations called for the services of an expert.....When viewed in this light the achievements of the unknown Hindu, who sometime in the first centuries of our era discovered the principle of position, assumes the importance of a world event."

Dantzig is puzzled at the fact that the great mathematicians of Greece did not stumble on this discovery.

"Is it that the Greeks had such a marked contempt for applied science, leaving even the instruction of their children to the slaves? But if so, how is it that the nation that gave us geometry and carried this science so far did not create a rudimentary algebra? that corner-stone of modern mathematics, also originated in India, and at about the same time that positional numeration did?"

(source: The Discovery of India - By Jawaharlal Nehru Oxford University Press. 1995 p. 218)

The Unsung Mathematician:

An important Mathematics book prescribed by the New York State Education Department acknowledges the debt in the following words:

"The Western world owes a great deal to India for a simple invention. It was developed by an unknown Indian more than 1500 years ago. Without it most of the great discoveries and inventions (including computers) of western civilization would never have come about. This invention was the decimal system of numerals - nine digits and a zero. The science and technology of today (including the computers) could not have developed if we had only the Roman system of numerals. That system is too clumsy to be used as a scientific too. Today we take the decimal system for granted. We don't think about how brilliant the man who invented zero must have been. Yet without zero we could not assign a place value to the digits. That ancient mathematician, whoever, he was, deserves much honor."

Indians also made advances in other areas of mathematics. Very early in their history they developed a simple system of geometry. This system was used to plan outdoor sites for Indian religious ceremonies. Indians also added to our knowledge of even more complicated branches of mathematics such as trigonometry and calculus. They studied these branches of mathematics in order to apply them to astronomy."

(source: Harry Shor and Gloria Meng, Exploring Algebra).

For more refer to The Infinitesimal Calculus: How and Why it Was Imported into Europe - By C. K. Raju and
Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibh?s? - By C. K. Raju.

Refer to Visualizing Indian heritage Digital Library Metaphor – By Nagnath R Ramdasi - CDAC.

Charles Seife, a journalist with Science magazine, has also written for New Scientist, Scientific American, The Economist, Science, Wired UK, The Sciences, and numerous other publications. He holds an M.S. in mathematics from Yale University and his areas of research include probability theory and artificial intelligence. He is a mathematician and science writer, author of Zero: The Biography of a Dangerous Idea says:

"Perhaps no one has embraced nothing as strongly as the Indians who, Seife notes, "never had a fear of the infinite or of the void." Hinduism has embedded within it, a complex philosophy of nothingness, seeing everything in the world as arising from the pregnant void, known as Sunya.

"The ultimate goal of the Hindu was to free himself from the endless cycle of pain found in continual reincarnation and reconnect with the Nothingness that is the source and fundament of the All. For Indians, the void of Sunya was the very font of all potential; nothingness was liberation. No surprise then that it is from this sophisticated culture that we inherit the mathematical analog of nothing, zero. Like Sunya, zero is a kind of place holder, a symbol signifying a pregnant space where any other number might potentially reside."

(source: Zero: The Biography of a Dangerous Idea: It's weird, it's counterintuitive and the Greeks hated it. Why did the Church reject the use of zero?
http://www.calendarlive.com/top/1,1419,L-LATimes-Books-X!ArticleDetail-26133,00.html
http://www.salon.com/books/review/2000/03/03/seife/index.html).

Lancelot Thomas Hogben (1895-1975) English zoologist and geneticist, has written:

"In the whole history of Mathematics, there has been no more revolutionary step than the one which the Hindus made when they invented the sign ‘0’ for the empty column of the counting frame."

(source: Mathematics for the Million - By Lancelot Thomas Hogben p. 47).

The concept of Debits and negative numbers originated in India, and why were they not accepted until recently? It was much more than 2000 years ago. It wasn't accepted elsewhere because the Church did not think it possible.

The paper of Reuben Burrow (1798-1868) "A Proof that the Hindus had the Binomial Theorem." (published in 1790) Asiatic Researches 2 (1790): 487-97 is more proof for us that the western world was aware of the Indian achievement in the field of combinational mathematics. Then, the problem would be one of explaining how the so called 'Pascal's triangle' continues to bear his name, or how the British reference books like the Encyclopedia Britannica persisted (till well into the 20th century) in crediting Newton with the discovery of the binomial theorem.

(source: India Through The Ages: History, Art Culture and Religion - By G. Kuppuram p. 672-673).

The Hindus knew mathematics much early. In the Rig Veda (2-18, 4 to 6), there are references to ‘two’, ‘four’, ‘eight’, ‘ten.’

Aa dvabhyam haribhyamindryahya
chaturbhirashadabhi rhuya manah ashtabhirdashabhih

Also in Vajasaneya Samhita (17.2), there is the passage referring to 1, 10, 100, 1000 etc.

Eka cha dasha cha dasha cha shatam cha shatam cha
sahasram cha sahasram cha yutam cha ayutam cha
niyutam cha niyutam cha prayutam cha. Etc.

In Mahabharata there are references to addition and subtraction. Adhikam (more), Unam (less), Shesham (remaining), multiplication and division are indicated. For example, “60 thousand camels and twice the number of horses” are referred to.

In Rig Veda (10.62.7), Nabhanedishta praises King Savarni for giving in charity one thousand cows, who had the figure 8 on their ears and so were called Ashta Karni. It seems that gambling was very common in the Vedic days, and it involved dices and numbers. According to Yajur Veda, Vajasneya Samhita (4.3,3), in the Rajasuya sacrifice, five was called Abhiburasi. In another kind of gambling, the dice (Aksha) used four names of the four Ages namely Krita, Treta, Dvapar and Kali and they were numbered 4, 3, 2 and 1. The numbers from one to one thousand billion are found in the Vajasneya Samhita and also in Taitteriya, Maitrayani and Kathaka Samhitas.

In Sama Veda, in the 25th Brahmana, there is a reference to how much fees (dakshina) should be given to a priest in sacrifice (Yajna). It may be at least 12 (Krishnala) milligrams of gold, and doubling the figure, it can go up to 3,93,216.

The system they adopt in giving page numbers in old manuscripts in Malabar and in Andhra was to have 34 digits of consonants from Ka to La and then to have the next 34 digits by adding vowels Kaa to Laa. They can number pages upto 408 (34 x 12). Burma also had the same system for pagination.

(source: Hinduism: Its Contribution to Science and Civilization - By Prabhakar Balvant Machwe p. 10 -14).

The Notion of Infinity and zero:

There is a beautiful definition of the infinite in the following line of a Vedic mantra, which forms the introductory verse to the Isa Upanishad:

It says: Take the whole (Infinite Brahman) from the whole, and the whole still remains. This is almost like the mathematician, Cantor's definition of infinity.

The very names of the numerals are of Sanskrit origin. Professor Arthur Macdonell says in his A History of Sanskrit Literature: "During the eighth and ninth centuries, the Indians became the teachers in arithmetic and algebra of the Arabs, and through them of the nations of the west. Thus, though we call the latter science by an Arabic name, it is a gift we owe to India."

(source: Indian Culture and the Modern Age - By Dewan Bahadur K. S. Ramaswami Sastri Annamalai University. 1956 p.66-67).

The linkage of God with the infinite is found in the Bhagavad Gita, by tradition spoken by Lord Krishna himself, we read:

“O Lord of the universe, I see You everywhere with infinite form…Neither do I see the beginning nor the middle nor the end of Your Universal Form.”

(source: Infinity: The Quest to Think the Unthinkable - By Brian Clegg p. 54).

Zero to Infinity in Indian Mysticism

Ananta is Sanskrit for infinity. It is equated with the Supreme Brahman — infinitely powerful and so infinitely free. It is bigger than any quantity that can be imagined; it is bigger than any finite number. Infinity is one of the fundamental axioms upon which contemporary mathematics is based. Sanskrit grammar and interpretation in ancient India were closely linked to the handling of high value numbers. Studies relating to poetry and metrics initiated sastragnaas or scientists to both arithmetic and grammar. Grammarians were just as competent at calculations as professional mathematicians. Indian sastragnaas or scientists, philosophers, astronomers and cosmographers — in order to develop their arithmetical, metaphysical and cosmological speculations concerning ever higher numbers — became at once mathematicians, grammarians and poets. They gave their spoken counting system a truly mathematical structure which had the potential to lead directly to the discovery of the decimal place-value system.

In Indian mysticism, the concept of infinity and zero are very closely linked. In the Isavasya Upanishad, there’s a line: “Poornasya poornam aadaya poornameva visish-yate”. To mathematically explain this, we have to assume that the first poornam represents infinity and the second, zero. In Sanskrit, poornam means both full and zero. Indian mathematicians knew perfectly well how to distinguish between these two notions which are mutually contradictory and which are the inverse of each other. They knew that division by zero gave them infinity. The concept of infinity has always remained an enigma. The Taittiriya Upanishad says: yatho vacho nivartante, apraapya manasa saha — where mind and speech return (being) unable to comprehend. In Indian cosmology, Ananta refers to the Adisesha or the great serpent on which Lord Vishnu reclines, taking His yoga nidra or anantasayanam.

The symbol for infinity is called the leminiscate. English mathematician John Wallis introduced this symbol for the first time in 1655. Hindu mythological iconography contains a similar symbol representing the same idea. The symbol is that of Ananta, the great Adisesha of infinity and eternity, which is always represented, coiled up in a horizontal figure of 8 just like the leminiscate.

Negative numbers had been rejected as solutions of problems in early times. They were eventually admitted in Hindu practical mathematics through problems involving money transactions, since the idea of receiving and owing money was a simple and obvious one — a negative number could be interpreted as a debt. Objection to negative numbers continued up to the early 19th century. Negative numbers are the mirror image of positive numbers. The invention of Cartesian geometry brought the X, Y co-ordinates and numbers came to be represented on a graph. Today, the series of negative natural numbers go up to infinity.

(source: Zero to Infinity in Indian Mysticism - By T R Rajagopalan - Times of India).
‘Calculus is India ’s Gift to Europe ’

Dr. C.K. Raju (1954 - ) holds a Ph.D. from the Indian Statistical Institute. He taught mathematics for several years before playing a lead role in the C-DAC team which built Param: India ’s first parallel supercomputer. His earlier book ‘Time: Towards a Consistent Theory’ (Kluwer Academic, 1994) set out a new physics with a tilt in the arrow of time. He has been a Fellow of the Indian Institute of Advanced Study and is a Professor of Computer Applications.

He has has revealed how calculus, an Indian invention, was picked up by the Jesuit priests from Kerala in the second half of the 16th century and taken to Europe. This is how the Westerners got their calculus. Overtime, people forgot this link and the Europeans began to claim calculus as their own invention. This myth still persists despite calculus texts existing in India since thousands of years.

“Indian infinite series has been known to British scholars since at least 1832, but no scholar tried to establish the connection with the calculus attributed to Newton and Leibnitz,” he said.

Dr. Raju’s 10-year research that included archival work in Kerala and Rome was published in a book “Cultural Foundations of Mathematics.” It established that the Jesuit priests took trigonometric tables and planetary models from the Kerala mathematicians of the Aryabhata school and exported them to Europe starting around 1560 in connection with the European navigational problem.

“When the Europeans received the Indian calculus, they couldn’t understand it properly because the Indian philosophy of mathematics is different from the Western philosophy of mathematics. It took them about 300 years to fully comprehend its working. The calculus was used by Newton to develop his laws of physics,”

It is well known that the “Taylor-series” expansion, that is at the heart of calculus, existed in India in widely distributed mathematics / astronomy / timekeeping (“jyotisa”) texts which preceded Newton and Leibniz by centuries.

Why were these texts imported into Europe ? These texts, and the accompanying precise sine values computed using the series expansions, were useful for the science that was at that time most critical to Europe : navigation. The ‘jyotisa’ texts were specifically needed by Europeans for the problem of determining the three “ells”: latitude, loxodrome, and longitude.

How were these Indian texts imported into Europe ? Jesuit records show that they sought out these texts as inputs to the Gregorian calendar reform. This reform was needed to solve the ‘latitude problem’ of European navigation. The Jesuits were equipped with the knowledge of local languages as well as mathematics and astronomy that were required to understand these Indian texts.

The Jesuits also needed these texts to understand the local customs and how the dates of traditional festivals were fixed by Indians using the local calendar (“panch?nga”). How the mathematics given in these Indian ancient texts subsequently diffused into Europe (e.g. through clearing houses like Mersenne and the works of Cavalieri, Fermat, Pascal, Wallis, Gregory, etc.) is yet another story.

The calculus has played a key role in the development of the sciences, starting from the “Newtonian Revolution”. According to the “standard” story, the calculus was invented independently by Leibniz and Newton . This story of indigenous development, ab initio, is now beginning to totter, like the story of the “Copernican Revolution”.

The English-speaking world has known for over one and a half centuries that “Taylor series” expansions for sine, cosine and arctangent functions were found in Indian mathematics / astronomy / timekeeping (‘jyotisa’) texts, and specifically in the works of Madhava, Neelkantha, Jyeshtadeva, etc. No one else, however, has so far studied the connection of these Indian developments to European mathematics.

The connection is provided by the requirements of the European navigational problem, the foremost problem of the time in Europe . Columbus and Vasco da Gama used dead reckoning and were ignorant of celestial navigation. Navigation, however, was both strategically and economically the key to the prosperity of Europe of that time.

Jesuits such as Matteo Ricci who trained in mathematics and astronomy under Clavius’ new syllabus were sent to India . In a 1581 letter, Ricci explicitly acknowledged that he was trying to understand the local methods of time-keeping (‘jyotisa’) from the Brahmins and Moors in the vicinity of Cochin .
Cochin was then the key centre for mathematics and astronomy since the Vijaynagar Empire had sheltered it from the continuous onslaughts of Islamic raiders from the north. Language was not a problem for the Jesuits since they had established a substantial presence in India .

For more refer to The Infinitesimal Calculus: How and Why it Was Imported into Europe - By C. K. Raju and
Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibh?s? - By C. K. Raju.

(source: ‘Calculus is India ’s Gift to Europe ’ - By Dr. C K Raju - indianrealist.com).

In his speech introducing the Indian Budget March 1st, 1926, Sir Basil Blackett said:

"India long ago revolutionized mathematics, and provided the West with the key to the most far reaching of all the mechanical instrument on which its control of nature has been built, when it presented to Europe through the medium of Arabia the device of the cypher (and the decimal notation) upon which all modern system of numeration depend. even so, India today or tomorrow, will, I am confident, revolutionize western doctrines of progress by demonstrating the insufficiency and lack of finality of much of the West's present system of human values."

(source: India in Bondage: Her Right to Freedom - Rev. Jabez T. Sunderland p.356-357).

Georges Ifrah ( ? ) French historian of Mathematics and author of the book, The Universal History of Numbers has written:

"The Indian mind has always had for calculations and the handling of numbers an extraordinary inclination, ease and power, such as no other civilization in history ever possessed to the same degree. So much so that Indian culture regarded the science of numbers as the noblest of its arts...A thousand years ahead of Europeans, Indian savants knew that the zero and infinity were mutually inverse notions."

(source: Histoire Universelle des Chiffres - By Georges Ifrah Paris - Robert Laffont, 1994, volume 2. p. 3).

“The real inventors of [the numeral system], which is no less important than such feats as the mastery of fire, the development of agriculture, or the invention of the wheel, writing or the steam engine, were the mathematicians and astronomers of Indian civilization: scholars who, unlike the Greeks, were concerned with practical applications and who were motivated by a kind of passion for both numbers and numerical calculations.”

Claiming India to be the true birthplace of our numerals, Ifrah salutes the Indian researchers saying that the "...real inventors of this fundamental discovery, which is no less important than such feats as the mastery of fire, the development of agriculture, or the invention of the wheel, writing or the steam engine, were the mathematicians and astronomers of the Indian civilization: scholars who, unlike the Greeks, were concerned with practical applications and who were motivated by a kind of passion for both numbers and numerical calculations."

He refers to 24 evidences from scriptures from India, whose dates range from 1150 BC until 458 BC. Of particular interest is the work by Indian mathematician Bhaskaracharya known as Bhaskara (1150 BC) where he makes a reference to zero and the place-value system were invented by the god Brahma. In other words, these notions were so well established in Indian thought and tradition that at this time they were considered to have always been used by humans, and thus to have constituted a "revelation" of the divinities.

"It was only after the eighth century BC, and doubtless due to the influence of the Indian Buddhist missionaries, that Chinese mathematicians introduced the use of zero in the form of a little circle or dot (signs that originated in India),...".

The early passion which Indian civilization had for high numbers was a significant factor contributing to the discovery of the place-value system, and not only offered the Indians the incentive to go beyond the "calculable" physical world, but also led to an understanding (much earlier than in our civilization) of the notion of mathematical infinity itself.

Sanskrit notation had an excellent conceptual quality. It was easy to use and moreover it facilitated the conception of the highest imaginable numbers. This is why it was so well suited to the most exuberant numerical or arithmetical-cosmogonic speculations of Indian culture."

"The Indian people were the only civilization to take the decisive step towards the perfection of numerical notation. We owe the discovery of modern numeration and the elaboration of the very foundations of written calculations to India alone."

"It is clear how much we owe to this brilliant civilization, and not only in the field of arithmetic; by opening the way to the generalization of the concept of the number, the Indian scholars enabled the rapid development of mathematics and exact sciences. The discoveries of these men doubtless required much time and imagination, and above all a great ability for abstract thinking. These major discoveries took place within an environment which was at once mystical, philosophical, religious, cosmological, mythological and metaphysical."

"In India, an aptitude for the study of numbers and arithmetical research was often combined with a surprising tendency towards metaphysical abstractions; in fact, the latter is so deeply ingrained in Indian thought and tradition that one meets it in all fields of study, from the most advanced mathematical ideas to disciplines completely unrelated to 'exact sciences.

In short, Indian science was born out of a mystical and religious culture and the etymology of the Sanskrit words used to describe numbers and the science of numbers bears witness to this fact. "

"Sanskrit means “complete”, “perfect” and “definitive”. In fact, this language is extremely elaborate, almost artificial, and is capable of describing multiple levels of meditation, states of consciousness and psychic, spiritual and even intellectual processes. As for vocabulary, its richness is considerable and highly diversified. Sanskrit has for centuries lent itself admirably to the diverse rules of prosody and versification. Thus we can see why poetry has played such a preponderant role in all of Indian culture and Sanskrit literature. "

(source: The Universal History of Numbers - By Georges Ifrah p 365 - 441).

Marcus du Sautoy (1965 - ) is a Professor of Mathematics at University of Oxford. Formerly of All Souls College, he is now a fellow of Wadham College. He has been named by The Independent on Sunday as one of the UK 's leading scientists. In 2001 he won the prestigious Berwick Prize of the London Mathematical Society, which is awarded every two years to reward the best mathematical research by a mathematician under forty.

In The Story of Maths, he says Indians made many of these breakthroughs before Newton was born.

The Story of Maths, a four-part series, will be screened on BBC Four in 2008. The first part looks at the development of maths in ancient Greece , ancient Egypt and Babylon ; the second focuses on India, China and Central Asia and the rest look at how maths developed in the West. The India reel focuses on how several Indians developed theories in maths that were later discovered by Westerners who took credit for them.

Aryabhatta (476–550 AD), who calculated pi, and Brahmagupta (598-670 AD) feature in the film, which also showcases a Gwalior temple, which documents the first inscription of 'zero'. "One of the biggest inventions in India was the number zero. Indians used it long before the West did," said Du Sautoy. "When the West had Roman numerals there was no zero and that is why they were so clumsy. On the other hand, Brahmagupta was one of the key mathematicians in the world because he invented the idea of zero."

The documentary also features the history of Kerala-born mathematician Madhava (1350-1425) who created calculus 300 years before Newton and German mathematician Gottfried Leibniz did, said Du Sautoy. "We learn that Newton invented the mathematical theory calculus in the 17th century but Madhava created it earlier," Du Sautoy said.

(source: Oxford prof documents India's math contribution – By Naomi Canton, Hindustan Times July 5, 2007).
Not Newton, but Madhava!

Subsets of calculus existed in the Ganita-Yukti-Bhasa two centuries before Isaac Newton published his work, according to a recently published translation of the manuscript.
Prof K Ramasubramanian of IIT-Bombay has recently released two-volume translation of the Ganita-Yukti-Bhasa by Jyesthdeva points to the fact that some subsets of calculus existed in Indian manuscripts almost two centuries before Isaac Newton published his work.

And that an Indian mathematician and astronomer Nilakantha Somayaji spoke, in parts, about a planetary model, credited to Tycho Brahe almost a century later.

In the Tantra Sangraha (The Tantra Sangraha is a treatise on astronomy and related mathematics in elegant verse form, in Sanskrit. It consists of 432 verses.) Nilakantha talks about a planetary model where five planets, which can be seen with the naked eye – Mercury, Venus, Mars, Jupiter and Saturn – move around the sun, which in turn moves around the earth.

The fact remains that a century later, Tycho Brahe published the same planetary model and was credited for it, since no one knew of Nilakantha’s work.

(source: Not Newton, but Madhava! - mumbaimmirror.com).

Dr George Gheverghese Joseph from the University of Manchester and author of best-selling book 'The Crest of the Peacock: the Non-European Roots of Mathematics' said the ' Kerala School ' identified the 'infinite series 'one of the basic components of calculus- -way back in AD1350.

"The beginnings of modern maths is usually seen as a European achievement, but the discoveries in medieval India between the 14th and 16th centuries have been ignored or forgotten."

"The brilliance of Newton 's work at the end of the 17th century stands undiminished - especially when it came to algorithms of calculus. But other names from the Kerala School, notably Madhava and Nilakantha, should stand shoulder to shoulder with him as they discovered the other great component of calculus- infinite series."

The discovery is now attributed in math books to British scientist Sir Isaac Newton and his German contemporary Gottfried Leibnitz at the end of the 17th century. There was strong circumstantial evidence that the Indians passed on their discoveries to mathematically knowledgeable Jesuit missionaries who visited India during the 15th century.

That knowledge may have eventually been passed on to Newton himself, he said.

"It's hard to imagine that the West would abandon a 500-year-old tradition of importing knowledge and books from India and the Islamic world. But we've found evidence there was plenty of opportunity to collect the information as European Jesuits were present in the area at that time."

"They were learned with a strong background in maths and were well versed in the local languages. And there was strong motivation: Pope Gregory XIII set up a committee to look into modernising the Julian calendar. On the committee was the German Jesuit astronomer/mathematician Clavius, who repeatedly requested information on how people constructed calendars in other parts of the world. The Kerala School was undoubtedly a leading light in this area. "Similarly there was a rising need for better navigational methods including keeping accurate time on voyages of exploration and large prizes were offered to mathematicians who specialised in astronomy.

"There were many such requests for information across the world from leading Jesuit researchers in Europe . Kerala mathematicians were hugely skilled in this area." There were many reasons why the contribution of the Kerala School has not been acknowledged till now. A prime reason, was the "neglect of scientific ideas emanating from the Non-European world - a legacy of European colonialism and beyond".

"But there is also little knowledge of the medieval form of the local language of Kerala, Malayalam, in which some of most seminal texts, such as the Yuktibhasa, from much of the documentation of this remarkable mathematics is written." "For some unfathomable reasons, the standard of evidence required to claim transmission of knowledge from East to West is greater than the standard of evidence required to knowledge from West to East." The Kerala School also discovered what amounted to the Pi series and used it to calculate Pi correct to 9, 10 and later 17 decimal places.

(source: When Kerala scholars beat Newton - rediff.com and Indians Predated Newton 'Discovery' By 250 Years, Scholars Say - sciencedaily.com).

Brian Clegg ( ? ) author of popular science books has written:

"The characters we use for the numbers arrived here from India via the Arabic world. The Brahmi numerals that have been found in caves and on coins around Mumbai from around the first century AD use horizontal lines for 1 to 3. The squiggles used for 4 to 9, however, are clear ancestors of the numbers we use today. These symbols were gradually taken up by Arabs and came to Western attention in the 13th century thanks to two books, on written by a traveler from Pisa, the other by a philosopher in Baghdad. The earlier book was written by the philosopher al-Khwarizmi in the 9th century. The Latin translation Algoritmi de numero Indorum (al-Khwarizmi on the numbers of the Hindus).

The translation of De numero Indorum slightly predates the man who is credited with introducing the system to the West. Leonardo of Pisa, or by his nickname Fibonacci. In the comments in his book Liberabaci, written in 1202, he states that he was “introduced to the art of Indian’s nine symbols” and it was this book that really brought the Hindu system to the West.

(source: Infinity: The Quest to Think the Unthinkable - By Brian Clegg p. 54 - 60).

Carl B. Boyer (1906 – 1976) in his "History of Mathematics" pages 227-228”. “...Mohammed ibn-Musa al-Khwarizmi, ..., who died sometime before 850, wrote more than a half dozen astronomical and mathematical works, of which the earliest were probably based on the Sindhind derived from India. Besides ... [he] wrote two books on arithmetic and algebra which played very important roles in the history of mathematics. ... In this work, based presumably on an Arabic translation of Brahmagupta, al-Khwarizmi gave so full an account of the Hindu numerals that he probably is responsible for the widespread but false impression that our system of numeration is Arabic in origin. ...
Edward Sachau, In a translation of Alberuni ‘s “Indica”, a seminal work of this period (c.1030 AD), writes this in his introduction, “Many Arab authors took up the subjects communicated to them by the Hindus and worked them out in original compositions , commentaries and extracts. A favourite subject of theirs was Indian mathematics..." etc.

“ Al-Khwarizmi wrote numerous books that played important roles in arithematic and algebra. In his work, De numero indorum (Concerning the Hindu Art of Reckoning), it was based presumably on an Arabic translation of Brahmagupta where he gave a full account of the Hindu numerals which was the first to expound the system with its digits 0,1,2,3,....,9 and decimal place value which was a fairly recent arrival from India. Because of this book with the Latin translations made a false inquiry that our system of numeration is arabic in origin. The new notation came to be known as that of al-Khwarizmi, or more carelessly, algorismi; ultimately the scheme of numeration making use of the Hindu numerals came to be called simply algorism or algorithm, a word that, originally derived from the name al-Khwarizmi, now means, more generally, any peculiar rule of procedure or operation.

Interestingly, as the article notes, “The Hindu numerals like much new mathematics were not welcomed by all. In 1299 there was a law in the commercial center of Florence forbidding their use; to this day this law is respected when we write the amount on a check in longhand .”

“It is now universally accepted that our decimal numbers derive from forms, which were invented in India and transmitted via Arab culture to Europe, undergoing a number of changes on the way. We also know that several different ways of writing numbers evolved in India before it became possible for existing decimal numerals to be marred with the place-value principle of the Babylonians to give birth to the system which eventually became the one which we use today. Because of lack of authentic records, very little is known of the development of ancient Hindu mathematics. The earliest history is preserved in the 5000-year-old ruins of a city at Mohenjo Daro, located Northeast of present-day Karachi in Pakistan. Evidence of wide streets, brick dwellings an apartment houses with tiled bathrooms, covered city drains, and community swimming pools indicates a civilisation as advanced as that found anywhere else in the ancient Orient.



These early peoples had systems of writing, counting, weighing, and measuring, and they dug canals for irrigation. All this required basic mathematics and engineering. “The special interest of the Indian system is that it is the earliest form of the one, which we use today. Two and three were represented by repetitions of the horizontal stroke for one. There were distinct symbols for four to nine and also for ten and multiples of ten up to ninety, and for hundred and thousand.”



“…Knowledge of the Hindu system spread through the Arab world, reaching the Arabs of the West in Spain before the end of the tenth century. The earliest European manuscript, which came from the Hindu numerals were modified in north-Spain from the year 976.” And finally an important point for those who maintain that the concept of zero was also evident in some other civilisations: “Only the Hindus within the context of Indo-European civilisations have consistently used zero.”

(source: Hindu contribution to Mathematics - By B Shantanu - indiacause.com). Watch Carl Sagan and Hindu cosmology – video

David Mumford, the eminent mathematician writes in his review of the book, Mathematics in India by Kim Plofker:

"Did you know that Vedic priests were using the so-called Pythagorean theorem to construct their fire altars in 800 BCE?; that the differential equation for the sine function, infinite difference form, was described by Indian mathematician-astronomers in the fifth century CE?; and that 'Gregory's' series PI/4 = 1 -1/3+1/3-… was proven using the power series for arctangent and, with ingenious summation methods, used to accurately compute PI in southwest India in the fourteenth century?

(source: Mathematics in India - Reviewed by David Mumford - AMS American Mathematical Society - Volume 57 Number 3).

Gopala and Hemachandra and rhythmic patterns

Donald Knuth (1938 - ) of Stanford University in The Art of Computer Programming also wrote about this:

"Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes. The number of such rhythms having n beats altogether is Fn+1; therefore both Gopala (before 1135) and Hemachandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21, ... explicitly."

The system that Fibonacci introduced into Europe came from India and used the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 with, most importantly, a symbol for zero 0.

(source: Who was Fibonacci? and Origins of Fibonacci number and Fibonacci numbers or Hemecandra numbers? and Gopala and Hemachandra numbers everywhere - sepiamutiny.com Hemachandra).

Ian G. Pearce ( ? ) has written: “Mathematics has long been considered an invention of European scholars, as a result of which the contributions of non-European countries have been severely neglected in histories of mathematics. Worse still, many key mathematical developments have been wrongly attributed to scholars of European origin. This has led to so-called Eurocentrism. ...The purpose of my project is to highlight the major mathematical contributions of Indian scholars and further to emphasize where neglect has occurred and hence elucidate why the Eurocentric ideal is an injustice and in some cases complete fabrication.”

“It is through the works of Vedic religion that we gain the first literary evidence of Indian culture and hence mathematics. Written in Vedic Sanskrit the Vedic works, Vedas and Vedangas (and later Sulbasutras) are primarily religious in content, but embody a large amount of astronomical knowledge and hence a significant knowledge of mathematics. ... 'The need to determine the correct times for Vedic ceremonies and the accurate construction of altars led to the development of astronomy and geometry.'”

“I feel it important not to be controversial or sweeping, but it is likely European scholars are resistant due to the way in which the inclusion of non-European, including Indian, contributions shakes up views that have been held for hundreds of years, and challenges the very foundations of the Eurocentric ideology. ... It is almost more in the realms of psychology and culture that we argue about the effect the discoveries of non-European science may have had on the 'psyche' of European scholars. ... To summarize, the main reasons for the neglect of Indian mathematics seem to be religious, cultural and psychological”

(source: Indian Mathematics: Redressing the balance' - 'Abstract' - By Ian G. Pearce – '(IGP-IM:RB) 'Mathematics in the service of religion: I. Vedas and Vedangas' and Conclusion.

For more refer to The Infinitesimal Calculus: How and Why it Was Imported into Europe - By C. K. Raju and
Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibh?s? - By C. K. Raju

Remarking on this valuable contribution specially the discovery of number from one to nine and zero, which is considered to be the greatest and the most important, next only to the introduction of letters, Prof. Halsted of USA holds that no discovery in Arithmetic has contributed so much in the development of human intelligence and power. The Hindus can claim to be superior to the Greeks for the introduction of this system.

(source: Ancient Indian Culture At A Glance - By Swami Tattwananda Calcutta, Oxford Book Co. 1962 p. 121).

Zero is the embodiment of purna (full), lopa (absence), akasa (universe), bindu (dot), sunya (circle), in Indian literary and cultural traditions. The concept got concretized in the form of a symbol like dot or circle to fill up the empty space created in Indian decimal place-value concept. The scientific advances of the West would have been impossible had scientists continued to depend upon the Roman numerals and been deprived of the simplicity and flexibility of the decimal system and its main glory, the zero.

A 10th century traveler Masaudi, in his Arabic work Meadows of Gold, records that a Hindu Raja called Pandit who counted nine digits by memory. Abu Zafar Muhammad Al Khwarizm also mentions Hindu mathematicians, as does Al Beruni. In the Journal of the Bengal Asiatic Society (1907 p. 475), Feroz Abadi is quoted to have given the history of ‘Hindsa’ (= 0).
The number ‘10’ is a special contribution of Hindu arithmetic. So the zero was called ‘Hindsa’ in Persian.

(source: Hinduism: Its Contribution to Science and Civilization - By Prabhakar Balvant Machwe p. 10-14).

Muhammad ibn Musa al-Khawarazmi 772-773 A.D. who journeyed east to India to learn the sciences of that time. He introduced Hindu numerals, including the concept of zero, into the Arab world. Abu Abdulla Muhammad Ibrahim-al-Fazari translated Sidhanta from Sanskrit into Arabic, which, according to George Sarton (1884-1956) the great Harvard historian of science, wrote in his monumental Introduction to the History of Science, provided "possibly the vehicle by means of which the Hindu numerals were transmitted from India to Islam".

 

Algebra

Brahmagupta gives the following rules concerning operations carried out on what he calls “fortunes” (dhana), “debts” (rina) and “nothing” (kha).

A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero (shunya) minus zero is nothing. (kha).
A debt subtracted from zero is a fortune.
So a fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multiplied by itself is nothing.
The product or the quotient of two fortunes is one fortune.
The product or the quotient of two debts is one debt.
The product or the quotient of a debt multiplied by a fortune is a debt.
The product or the quotient of a fortune multiplied by a debt is a debt.

Modern algebra was born, and the mathematician had thus formulated the basic rules: by replacing “fortune” and “debt” respectively with “positive number” and “negative number”, we can see that at that time the Indian mathematicians knew the famous “rule of signs” as well as all the fundamental rules of algebra.

(source: The Universal History of Numbers - By Georges Ifrah p 439).

Florian Cajori (1859 - 1930) Swiss-born U.S. educator and mathematician whose works on the history of mathematics says:

"Indians were the “real inventors of Algebra”

(source: Is India Civilized - Essays on Indian Culture - By Sir John Woodroffe Ganesh & Co. Publishers 1922 p. 182).

Friedrich Rosen (1805-1837) edited and translated in 1831, The Algebra of Mohammed ben Musa. This is the oldest Arabic on mathematics and it shows that the Arabs borrowed algebra from India.

(source: German Indologists: Biographies of Scholars in Indian Studies writing in German - By Valentine Stache-Rosen p.24-25).

Algebra went to Western Europe from the Arabs - i.e. (Al-jabr, adjustment) who adopted it from India rather than from Greece. Sir Monier-Williams, T. S. Colebrooke, and Macdonell hold that the Arabs got Algebra from the Hindus. The great Indian leaders in this field, as in astronomy were Aryabhata, Brahmagupta, and Bhaskara. The last appears to have invented the radical sign and many algebraic symbols. These men created the conception of a negative quantity, without which algebra would have been impossible; they found the square root of 2, and solved, in the eighth century A.D., indeterminate equations of the second degree that were unknown to Europe until the days of Euler a thousand years later. They expressed their science in poetic form and gave to mathematical problems a grace characteristic to India's Golden Age.

Henry Thomas Colebrooke (1765-1837) wrote: "They (the Hindus) understood well the arithmetic of surd roots; they were aware of the infinite quotient resulting from the division of finite quantities by cipher; they knew the general resolution of equations of the second degree, and had touched upon those of higher denomination, resolving them in the simplest cases, and in those in which the solution happens to be practicable by the method which serves for quadratics; they had attained a general solution of indeterminate problems of the first degree; they had arrived at a method for deriving a multitude of solutions or answers to problems of the second degree from a single answer found tentatively."

"And this, says Colebrooke in conclusion, was as near an approach to a general solution of such problems as was made until the days of La Grange."

(source: Miscellaneous Essays - By H. T. Colebrooke Volume II p. 416 - 418).


" Out of a swarm of bees one-fifth part settled on a Kadamba blossom; one-third on a Silindhra flower; three times the difference of those numbers flew to the bloom of a Kutaja. One bee, which remained, hovered about in the air. Tell me, charming woman, the number of bees ...Eight rubies, ten emeralds, and a hundred pearls, which are in thy ear-ring, my beloved, were purchased by me for thee at an equal amount; and the sum of the prices of the three sorts of gem was three less than half a hundred; tell me the price of each, auspicious woman."

 

"The Indian mind has always had for calculations and the handling of numbers an extraordinary inclination, ease and power, such as no other civilization in history ever possessed to the same degree. So much so that Indian culture regarded the science of numbers as the noblest of its arts."

Refer to Indian Institute of Scientific Heritage


Aryabhata (475 A.D. - 550 A.D.) is the first well known Indian mathematician. Born in Kerala, he completed his studies at the university of Nalanda. In the section Ganita (calculations) of his astronomical treatise Aryabhatiya (499 A.D.) he made the fundamental advance in finding the lengths of chords of circles, by using the half chord rather than the full chord method used by Greeks. He gave the value of pi as 3.1416, claiming, for the first time, that it was an approximation. (He gave it in the form that the approximate circumference of a circle of diameter 20000 is 62832.) He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations of the type ax - by = c, and also gave what later came to be known as the table of Sines. He also wrote a text book for astronomical calculations, Aryabhatasiddhanta. Even today, this data is used in preparing Hindu calendars (Panchangs). In recognition to his contributions to astronomy and mathematics, India's first satellite was named Aryabhata.




Aryabhatta (475 A.D. - 550 A.D).

Aryabhatta put forward a brilliant thesis with regard to the Earth's rotation on its axis.

(image source: Vishwa Hindu Parishad of America. Inc - 2002 calendar).

 

Soviet historians, K. Antonova, G. Bongard-Levin, and G. Kotovsky, authors of A History of India, Moscow, Volume I and II 1973, have spoken highly of scientists of ancient India and their high originality:

"In the ancient period and in the early Middle Ages lived the outstanding mathematicians Aryabhatta (5-6th centuries), Varahamihira (6th century) and Brahmagupta (late 6th and early 7th centuries), whose discoveries anticipated many scientific achievements of modern times. Aryabhata knew that pi equaled 3.1416. The theorem known to us as Pythagoras' theorem was also known at that time. Aryabhata proposed an original solution in whole numbers to the linear equations with two unknowns that closely resembles modern solutions.

"The ancient Indians evolved a system for calculation using zero, which was later taken over by the Arabs (the so-called Arabic numerals) and alter from them by other peoples. The Aryabhatta school was also familiar with sine and cosine.

"Scholars of the Gupta period were already acquainted with the movement of the heavenly bodies, the reasons for eclipses of the Sun and the Moon. Aryabhatta put forward a brilliant thesis with regard to the Earth's rotation on its axis."

"Aryabhatta's follower, Brahmagupta, put forward solutions for a whole series of equations."

"Indian scholars of this period also scored important successes in the sphere of astronomy. Certain astronomical treatises of this period have been preserved, and these siddhantas bear witness to the high level of astronomical knowledge attained by the ancient Indians."

"Brahmagupta (many centuries before Newton) suggested that objects fall to the ground as a result of terrestrial gravity."

"Interesting material relating to astronomy, geography and mineralogy is found in Varahamihira's work Brihat-samhita...."

(source: A History of India - By K. Antonova, G. Bongard-Levin, and G. Kotovsky Moscow, Volume I and II 1973 p. 169-171).

Aryabhatta was a great astronomer of remarkable originality. He is famous for his suggestions of the diurnal revolution of the earth on its own axis. Another important conclusion was about the apparent motion of the sun and the moon. He observes: "The starry vault is fixed: it is the earth which, moving on its own axis, seems to cause the rising and the setting of the planets and stars."

(source: Main Currents in Indian Culture - By S. Natarajan - The Institute of Indo-Middle East Cultural Studies. 1960. p 62-63).

Yavadvipa, the ancient name for Java, to which Sugriva sent search parties looking for Sita, is a Sanskrit name mentioned in the Ramayana. Aryabhatta wrote that when the sun rose in Sri Lanka, it was midday in Yavakoti (Java) and midnight in the Roman land. In the Surya Siddanata reference is also made to the Nagari Yavakoti with golden walls and gates.

(source: India and World Civilization - By D. P. Singhal Pan Macmillan Limited. 1993. p. 323).

Mnemonic and shorthand code letters were used by the Hindu astronomer Aryabhat, who composed his Aryabhatiya in 499 A.D. He answers the question: “How many times does the Earth rotate in a Mahayuga?” by the sutra – Ngishi Bunlrukshshru. Its letters count up to 15,82,23,75,200.

The second Aryabhatta (II) has also given such cryptic numberal-alphabets:

Kanadhajhajhujhila = 1599993
Mudayasinadha = 58179

(source: Hinduism: Its Contribution to Science and Civilization - By Prabhakar Balvant Machwe p. 10-14).

Comparing the Hindus and the Greeks as regards their knowledge of algebra, Sir Mountstuart Elphinstone says:

"There is no question of the superiority of the Hindus over their rivals in the perfection to which they brought the science. Not only is Aryabhatta superior to Diaphantus (as is shown by his knowledge of the resolution of equations involving several unknown quantities, and in general method of resolving all indeterminate problems of at least the first degree), but he and his successors press hard upon the discoveries of algebraists who lived almost in our own time!"

(source: History of India - By Mountstuart Elphinstone London: John Murray Date of Publication: 1849 p. 131).

The Aryabhatiya was translated into Latin in the 13th century. Through this translation European mathematicians eventually learned methods for calculating the squares of triangles and the volumes of spheres, as well as square and cube roots. He had conceptualized the ideas about the cause of eclipses and the sun being the source of moonlight a thousand years before the Europeans. A revolutionary thinker in many areas, Aryabhata gave the radius of the planetary orbits in terms of the radius of the earth-sun orbit – that is, their orbits as basically their periods of rotation around the sun. He explained that the glow of the moon and planets was the result of reflected sunlight. And with incredible astuteness, he conceptualized the orbits of the planets as ellipses, a thousand years before Kepler reluctantly (he originally preferred circles) came to the same conclusion. His value for the length of the year at 365 days, six hours, twelve minutes, and thirty seconds, however, is a slightly overestimate; the true value is fewer than 365 days and 6 hours.

"Brahmagupta became the head of the astronomical observatory at Ujjain, the foremost mathematical center of ancient India, where great mathematicians such as Varahamihira had worked and built a strong school of mathematical astronomy. The Brahmasphutasidhanta contains 25 chapters, the first ten of which are arranged by topics such as true longitudes of the planets, lunar eclipses, solar eclipses, rising and settings, the moon’s crescent, the moon’s shadow, conjunctions of the planets with the fixed stars. A large part of the Brahmasphutsidhanta was translated into Arabic in the early 770s and became the basis of various studies by the astronomer Ya’qub ibn Tariq. In 1126 it was translated into Latin. This translation, along with other associated texts translated from Arabic, provided the basis for the Indo-Arabic stage of Western astronomy. The culmination of southern Indian astronomy was the tradition begun by Madhava in Kerala right before 1400. Madhava was renowned for his derivation of the infinite series for pi and the power series for trigonometric functions. His pupil Paramesvara attempted to correct the lunar parameters by conducting a long series of eclipse observations between 1393 and 1432. In these observations he used an astrolabe, an instrument devised to measure the positions of heavenly bodies, to determine the angle of altitude of the eclipsed body and possibly, the time of the phase of the eclipses."

(source: Lost Discoveries: The Ancient Roots of Modern Science - By Dick Teresi p. 133 - 136).

For more refer to The Infinitesimal Calculus: How and Why it Was Imported into Europe - By C. K. Raju and
Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibh?s? - By C. K. Raju

In the Jewish Encyclopedia Vol. XII p 689, it is noted,

"Aryabhatta, the noted Hindu astronomer who lived about 476 A.D. and who is called the Newton of the country, wrote many works on Algebra and Geometry. He first discovered the rotation of the earth round its own axis. As a Jewish writer says the theory that earth is a sphere revolving round its own axis which immortalized Copernicus, was previously known to the Hindus, who were instructed in the truth of it by Aryabhatta."

Jogesh Chandar Roy (1859-1965) Eminent scholar, educationist, writer, linguist, historian. Owing to his talent was conferred many accolades like D.Litt., Acharya, Bidyanidhi, Roy Bahadur etc. He held that the Vedic sages first admitted that the world is round ohterwise the advent of dawn (Usha) in the hymns, before sunrise becomes meaningless."

(source: Ancient Indian Culture At A Glance - By Swami Tattwananda Calcutta, Oxford Book Co. 1962 p. 126).

Arya Bhatta (476 - 550 AD) Father of Astronomy, who in the fifth century AD that is over a millennium and a half ago very accurately calculated many aspects of the spinning earth.

Arya Bhatta"s seminal work Aryabhatiyam was translated in to Arabic in the ninth century and subsequently it journeyed further west. Four hundred years later in the thirteenth century this treatise on astronomy was translated into Latin. The Latin version of Aryabhatiyam provided the foundation for growth of astronomy in Europe . I want my readers to appreciate that the Hindu astronomers of India built the foundation for European astronomy to stand and flourish. Arya Bhatta in a chapter of his treatise entitled Gola, which means circle or sphere, very categorically demonstrates the sphericity of earth, way before any European astronomer even had the vaguest inkling as to the size and shape of our planet. Arya Bhatta"s calculation of the equatorial circumference of earth at 39,968 km is only marginally less (by 62 km) of 40,076 km that we accept today. His calculated duration of one complete rotation of earth round its axis, which in common parlance is stated as 1 day, is absolutely correct at 23 hours 56 minutes and 4.1 seconds. Arya Bhatta also calculated the value of Pi with remarkable accuracy, and laid the foundation of many disciplines of mathematics that included trigonometry and mensuration.

(source: Sea level rise and inundation of coastal India – By Nichiketa Das - indiacause.com).

Brahmagupta (598 A.D. - 665 A.D.) is renowned for introduction of negative numbers and operations on zero into arithmetic. His main work was Brahmasphutasiddhanta, which was a corrected version of old astronomical treatise Brahmasiddhanta. This work was later translated into Arabic as Sind Hind. He formulated the rule of three and proposed rules for the solution of quadratic and simultaneous equations. He was the first mathematician to treat algebra and arithmetic as two different branches of mathematics. He gave the solution of the indeterminate equation Nx2+1 = y2. He is also the founder of the branch of higher mathematics known as "Numerical Analysis".

The Hindus were aware of the length of diameter and circumference of the earth. According to Brahmagupta and Bhaskarachary the diameter is 7182 miles, some calculate it to be 7905 miles, modern scientists take it to be 7918 miles. For the sake of astronomical experiments the Hindus introduced Sanka Yantra and Ghati Yantra, the apparatus for measurement.

(source: Ancient Indian Culture At A Glance - By Swami Tattwananda Calcutta, Oxford Book Co. 1962 p. 126).

After Brahmagupta, the mathematician of some consequence was Sridhara, who wrote Patiganita Sara, a book on algebra, in 750 A.D. Even Bhaskara refers to his works. After Sridhara, the most celebrated mathematician was Mahaviracharaya or Mahavira. He wrote Ganita Sara Sangraha in 850 A.D., which is the first text book on arithmetic in present day form. He is the only Indian mathematician who has briefly referred to the ellipse (which he called Ayatvrit). The Greeks, by contrast, had studied conic sections in great detail.

Bhaskara (1114 A.D. - 1185 A.D.) or Bhaskaracharaya is the most well known ancient Indian mathematician. He was born in 1114 A.D. at Bijjada Bida (Bijapur, Karnataka) in the Sahyadari Hills. He was the first to declare that any number divided by zero is infinity and that the sum of any number and infinity is also infinity. He is famous for his book Siddhanta Siromani (1150 A.D.). It is divided into four sections - Leelavati (a book on arithmetic), Bijaganita (algebra), Goladhayaya (chapter on sphere - celestial globe), and Grahaganita (mathematics of the planets). Leelavati contains many interesting problems and was a very popular text book. Bhaskara introduced chakrawal, or the cyclic method, to solve algebraic equations. Six centuries later, European mathematicians like Galois, Euler and Lagrange rediscovered this method and called it "inverse cyclic". Bhaskara can also be called the founder of differential calculus. He gave an example of what is now called "differential coefficient" and the basic idea of what is now called "Rolle's theorem". Unfortunately, later Indian mathematicians did not take any notice of this. Five centuries later, Newton and Leibniz developed this subject. As an astronomer, Bhaskara is renowned for his concept of Tatkalikagati (instantaneous motion).

(source: Ancient Indian Mathematicians and http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bhaskara_II.html).

For more refer to The Infinitesimal Calculus: How and Why it Was Imported into Europe - By C. K. Raju and
Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibh?s? - By C. K. Raju

“A Persian translation of the Veeju-Ganitu was made in India,” says Mr. Edward Strachey, “in the year 1634, by Ata Oollah Rusidee.” The same gentlemen says, “Foizee, in 1587, translated the Leelavatee, a work on arithmetic, mensuration,” etc. from which work it appears that “Bhaskara must have written about the end of the 12th century..”

“We must not,” adds Edward Strachey author of Bija ganita; or, The algebra of the Hindus, “be too fastidious in our belief, because we have not found the works of the teachers of Pythagoras; we have access to the wreck only of their ancient learning; but when such traces of a more perfect state of knowledge; we see that the Hindoo algebra 600 years ago, had, in the most interesting parts, some of the most curious modern European discoveries, and when we see, that it was at that time applied to astronomy, we cannot reasonably doubt the originality and the antiquity of mathematical learning among the Hindoos.”

(source: A View of the History, Literature, and Mythology of the Hindoos - By William Ward (1769-1823) volume II p 329 London 1822).

Sir Mountstuart Elphinstone wrote: "In the Surya Siddhanta is contained a system of trigonometry which not only goes beyond anything known to the Greeks, but involves theorem which were not discovered in Europe till two centuries ago."

(source: Sanskrit Civilization - By G. R. Josyer p. 2).

The discovery of the law of gravitation which immortalized Newton was known in India by Bhaskaracharya long before the birth of Newton. In support of the assumption of this view there is sufficient evidence in a verse in Sidhanta Siromany by its author. Bhaskaracharya holds that when the earth which is endowed with the power of attraction drags with her own power heavy objects on the sky it appears that objects are falling but actually they are not falling, they are only being dragged by the power of attraction of the earth. When everything on the sky drags each other equally where will the earth fall: It is explained that earth, planets, stars, moon, sun etc - each of them is being dragged by the other with its respective power of attraction and as a result of this attraction none of them is removed from its axis.

(source: Ancient Indian Culture At A Glance - By Swami Tattwananda Calcutta, Oxford Book Co. 1962 p. 127).

Sir William Wilson Hunter wrote: "The Hindus attained a very high proficiency in arithmetic and algebra independently of any foreign influence." The romance of the composition of Lilavati - the standard Hindu text book on Arithmetic by Bhaskaracharya - is very interesting and charming. It deals not only with the basic elements of the science of arithmetic but also with questions of interest, of barter, of permutations and combinations, and of mensuration. Bhaskaracharya knew the law of gravitation. The Surya Siddhanta is based on a system of trigonometry. Professor Wallace says: "In fact it is founded on a geometrical theorem, which was not known to the geometricians of Europe before the time of Vieta, about two hundred years ago. And it employs the sine of arcs, a thing unknown to the Greeks." The 47th proposition of Book I of Euclid, which is ascribed to Pythagoras was known long ago to the Hindus and must have been learnt from them by Pythagoras.

(source: Indian Culture and the Modern Age - By Dewan Bahadur K. S. Ramaswami Sastri Annamalai University. 1956 p. 67).

For more refer to chapter on Greater India: Suvarnabhumi and Sacred Angkor

 

Geometry

Geometry, like Astronomy, owes its origin in India to religion, and Grammar and Philosophy too were similarly inspired by religion.

As George Frederick William Thibaut (1848-1914) author of Mathematics in the making in Ancient India, remarked: "The want of some rule by which to fix the right time for the religious altar gave the first impulse to astronomical observations; urged by this the priest, remained watching night after night the advancement of the moon through the circle of the Nakshatras...The laws of phonetics were investigated....the wrong pronunciation of a single letter of the text; grammar and etymology had the task of securing the right understanding of the holy texts. And Thibaut then lays down the principle, which should never be overlooked by Indian historians, that whatever science "is closely connected with the Ancient Indian religion, must be considered as having sprung up among the Indians themselves, and not borrowed from other nations."

Geometry was developed in India from the rules of the construction of the altars. The Black Yajur Veda (V.4.11) enumerates the different shapes in which altars could be constructed and Baudhayana and Apastamba furnish us with full particulars about the shape of these chitis and the bricks which had to be employed for their construction. The Sulva Sutras date from the eighth century before Christ. The geometrical theorem that the square of the hypotenuse is equal to the squares of the other two sides of a rectangular triangle is ascribed by the Greeks to Pythagoras; but it was known in India at least two centuries before, and Pythagoras undoubtedly learnt this rule from India.

(source: Journal of the Asiatic Society of Bengal, 1875. p. 227 and A History of Civilization in Ancient India Based on Sanscrit Literature - By Romesh Chunder Dutt p. 240-243)

Vedic altars and sacrificial places were constructed according to strict geometrical principles. The Vedic (altar) had to be stacked in a geometrical form with the sides in fixed proportions, and brick altars had to combine fixed dimensions with a fixed number of bricks. Again, the surface areas were so designed that altars could be increased in size without change of shape, which required considerable geometrical ingenuity.

Geometrical rules found in the Sulvasutras, therefore, refers to the construction of squares and rectangles, the relation of the diagonal to the sides, equivalent rectangles and squares, equivalent circles and squares, conversion, of oblongs into squares and vice versa, and the construction of squares equal to the sum or difference of two squares. In such relations a prior knowledge of the Pythagorean theorem, that the square of the hypotenuse of a right-angled triangle is equal to the sum of squares of the other two sides, is disclosed.

In measurement and construction of altars the priests formulated the Pythagorean theorem (by which the square of the hypotenuse of a right-angled triangle equals the sum of the squares of the other side) several hundred years before the birth of Christ.

As every schoolchild knows, the most important theorem in geometry is that of Pythagoras. Yet, there is no evidence that either the statement or the proof was known by the man to whom it is credited. The earliest statement can be found in the Sulbasutra of Baudhyana. Baudhayana has preserved its germination in religious rituals. The fact that ancient Indians knew this theorem was recognized quite early by some European scholars. Among the first was G. Thibaut, a historian of science, who left the impression that in geometry the Pythagoreans were the pupils of the Indians. Scholars unhappy with this idea tried to refute it, thought their refutation was, as Abraham Seidenberg, noted, were no more haughty dismissals.

The Formula known today as the Pythagorean Theorem was first postulated by Indian mathematician - Baudhayana in the 6th century C. E. long before Europe's math whizzies. In 497 C.E. Aryabhatta calculated the value of "pi" as 3.1416. Algebra, trigonometry and the concepts of algorithm, square root originated in India. Quadratic equations were propounded by Sridharacharya in the 11th century.

The largest number used by Greeks and Romans were 106, whereas Indians used numbers as big as 10 to the power of 53, as early as 5000 BCE. Even geometry called Rekha Ganita in ancient India, was applied to draft mandalas for architectural purposes and for creating temple motifs.

Professor H. G. Rawlinson writes: " It is more likely that Pythagoras was influenced by India than by Egypt. Almost all the theories, religions, philosophical and mathematical taught by the Pythagoreans, were known in India in the sixth century B.C., and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat and regarded certain vegetables such as beans as taboo" "It seems that the so-called Pythagorean theorem of the quadrature of the hypotenuse was already known to the Indians in the older Vedic times, and thus before Pythagoras

(source: Legacy of India 1937, p. 5).

Romesh Chunder Dutt, the famous Indian historian holds that the world is indebted to the Hindus for Geometry and not to the Greeks.

(source: Ancient Indian Culture At A Glance - By Swami Tattwananda Calcutta, Oxford Book Co. 1962 p. 124).

Professor Maurice Winternitz is of the same opinion: "As regards Pythagoras, it seems to me very probable that he became acquainted with Indian doctrines in Persia." (Visvabharati Quarterly Feb. 1937, p. 8).

It is also the view of Sir William Jones (Works, iii. 236), Colebrooke (Miscellaneous Essays, i. 436 ff.). Schroeder (Pythagoras und die Inder), Garbe (Philosophy of Ancient India, pp. 39 ff), Hopkins (Religions of India, p. 559 and 560) and Macdonell (Sanskrit Literature, p. 422).

(source: Eastern Religions & Western Thought - By S. Radhakrishnan ISBN: 0195624564 p. 143).

Ludwig von Schr?der German philosopher, author of the book Pythagoras und die Inder (Pythagoras and the Indians), published in 1884, he argued that Pythagoras had been influenced by the Samkhya school of thought, the most prominent branch of the Indic philosophy next to Vedanta.

(source: In Search of The Cradle of Civilization: : New Light on Ancient India - By Georg Feuerstein, Subhash Kak & David Frawley p. 252).

" Nearly all the philosophical and mathematical doctrines attributed to Pythagoras are derived from India."

Sir William Temple, (1628-1699) English statesman and diplomat, in his Essay upon the Ancient and Modern Learning (1690) he wrote:

"From these famous Indians, it seems most probable that Pythagoras learned, and transported into Greece and Italy, the greatest part of his natural and moral philosophy, rather than from the Aegyptians...Nor does it seem unlikely that the Aegyptians themselves might have drawn much of their learning from the Indians..long before..Lycurgus, who likewise traveled to India, brought from thence also the chief principles of his laws."

Temple's ideas remained in isolation in his period until they were revived in the middle of the 18th century when a battle raged between the 'believers' and the 'infidels' on the question of the value of Mosaic interpretation of history.

(source: Much Maligned Monsters: A History of European Reactions to Indian Art - By Partha Mitter p. 191).

Aryabhata, found the area of a triangle, a trapezium and a circle, and calculated the value of "pi" ( the relation of diameter to circumference in a circle) at 3.1416 - a figure not equaled in accuracy until the days of Purbach (1423-61) in Europe. Bhaskara anticipated the differential calculus, Aryabhata drew up a table of sines, and the Surya Siddhanta provided a system of trigonometry more advance than anything known to the Greeks. He had tabulated the sine function (unknown in Greece) for every 33/4? of arc from 33/4? to 90?. By 670 the system had reached northern Mesopotamia, where the Nestorian bishop Severus Sebokht praised its Hindu inventors as discoverers of things more ingenious than those of the Greeks. Muslims began the acquisition of foreign learning, and, by the time of the Caliph al-Mansur (d. 775), such Indian and Persian astronomical material as the Brahma-sphuta-siddhanta and the Shah's Tables had been translated into Arabic.

A 3,000-year-old ritual was resurrected at Panjal in Kerala in April 1975. A 12-day Agnicayana, or Atiratra, was performed on a bird-shaped altar of a thousand bricks. The altar was a geometricians' delight.

The area of each layer of the altar, for instance, was seven and a half times a square purusa, the size of the sacrificer or the Yajamana. A fifth of the size of the Yajamana, panchami, was the basic unit of the bricks.

The rules for measurement and construction of sacrificial altars are found in the Sulba Sutras, the earliest documents of geometry in India. Sulba means cord. Of the various Sulba Sutras, those of Baudhayana, Apastamba and Katyayana are best known. The mathematical knowledge in the texts comes from the creation of altars or bricks in various shapes-rhombus, isosceles trapezium, square, rectangle, isosceles right-angled triangle or circle. A square-shaped altar sometimes had to become circular without any change in the area or vice-versa. Obviously, the authors of the Sulba texts knew the value of pi, which is the ratio of the circumference to the diameter of a circle.

The theory of right angles is attributed to Greek philosopher Pythagoras (6th century BC). But Baudhayana mentions that the diagonal of a rectangle produces by itself both (the areas) produced separately by its two sides. In simple terms, this means that the square of the diagonal is equal to the sum of the squares of two sides. In the next rule he says that the rectangles for which the theorem is true have the sides as 3 and 4 [32+42=52], 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36. The theorem is given in all the Sulba Sutras.

Eminent mathematician A. K. Bag, he says tackling of mathematical and geometrical problems with rational numbers and irrational numbers [such as square-root of 2] was a unique achievement of early Indians. They even had technical terms such as dvikarani, trikarani and panchakarani (for square-roots of 2, 3 and 5) and so on and gave their values to a high degree of approximation.

The mathematics in Sulba texts also involves a highly sophisticated brick technology. Ten types of bricks were used to build the altar at Panjal.

Sir Monier-Williams says: "To the Hindus is due the invention of algebra and geometry, and their application to astronomy."

(source: Indian Wisdom - By Monier Williams p. 185).

Count Magnus Fredrik Ferdinand Bjornstjerna author of Theogony of the Hindus says: "We find in Ayeen-Akbari, a journal of the Emperor Akbar, that the Hindus of former times assumed the diameter of a circle to be to its periphery as 1,250 to 3,927. The ratio of 1,250 to 3,927 is a very close approximation to the quandrature of a circle, and differs very little form that given by Metius of 113 to 355. In order to obtain the result thus found by the Brahmans, even in the most elementary and simplest way, it is necessary to inscribe in a circle a poligon of 768 sides, an operation, which cannot be performed arithmetically without the knowledge of some peculiar properties of this curved line, and at least an extraction of the square root of the ninth power, each to ten places of decimals. The Greeks and Arabs have not given anything so approximate."

Professor Wallace says: "However ancient a book may be in which a system of trigonometry occurs, we may be assured it was not written in the infancy of the science. Geometry must have been known in India long before the writing of Surya Siddanata." which is supposed by the Europeans to have been written before 2000 B. C. E.

(source: Sanskrit Civilization - By G. R. Josyer p. 2-3).

Influence of Hindu Geometry on Greeks:

In his monumental work, The origin of mathematics, Archive for History of Exact Sciences. vol. 18, 301-342, Abraham Seidenberg remarks: "By examining the evidence in the Shatapatha Brahmana, we now know that Indian geometry predates Greek geometry by centuries. For example, the earth was represented by a circular altar and the heavens were represented by a square altar and the ritual consisted of converting the circle into a square of an identical area. There we see the beginnings of geometry! Two aspects of the 'Pythagoras' theorem are described in the Vedic literature. One aspect is purely algebraic that presents numbers a, b, c for which the sum of the squares of the first two equals the square of the third. The second is the geometric, according to which the sum of the areas of two square areas of different size is equal to another square. The Babylonians knew the algebraic aspect of this theorem as early as 1700 BCE, but they did not seem to know the geometric aspect. The Shatapatha Brahmana, which precedes the age of Pythagoras, knows both aspects. Therefore, the Indians could not have learnt it from the Old-Babylonians or the Greeks, who claim to have rediscovered the result only with Pythagoras. India is thus the cradle of the knowledge of geometry and mathematics."

So, contrary to the European belief that Hindus were influenced by the Greek geometry, the facts prove that it is the other way round. Most of the aspects of planar geometry described by Euclid and other Greek mathematicians were already known to Indians at least 2500 years before the Greeks. In fact, there are proofs which hint towards the fact Greeks were influenced by the ancient Hindu Mathematics and Geometry. Bibhuti Bhushan Datta in his book "Ancient Hindu Geometry" states:

"...One who was well versed in that science was called in ancient India as samkhyajna (the expert of numbers), parimanajna (the expert in measuring), sama-sutra-niranchaka (Uinform-rope-stretcher), Shulba-vid (the expert in Shulba) and Shulba-pariprcchaka (the inquirer into the Shulba). Of these term, viz, 'sama-sutra-niranchaka' perhaps deserves more particular notice. For we find an almost identical term, 'harpedonaptae' (rope-stretcher) appearing in the writings of the Greek Democritos (c. 440 BC). It seems to be an instance of Hindu influence on Greek geometry. For the idea in that Greek term is neither of the Greeks nor of their acknowledged teachers in the science of geometry, the Egyptians, but it is characteristically of Hindu origin." The English word 'Geometry' has a Greek root which itself is derived from the Sanskrit word 'Jyamiti'. In Sanskrit 'Jya' means an arc or curve and 'Miti' means correct perception or measurement.

The Sulba Sutras, however, date from about the eighth century B.C. E. and Dr. Thibault has shown that the geometrical theorem of the 47th proposition, Book I, which tradition ascribes to Pythagoras, was solved by the Hindus at least two centuries earlier, thus confirming the conclusion of Von Schroeder that the Greek philosopher owed his inspiration to India.

(source: History of Hindu Chemistry, Volume I p. XXIV ).
 

A. L. Basham, foremost authority on ancient India, writes in The Wonder That Was India:

"Medieval Indian mathematicians, such as Brahmagupta (seventh century), Mahavira (ninth century), and Bhaskara (twelfth century), made several discoveries which in Europe were not known until the Renaissance or later. They understood the import of positive and negative quantities, evolved sound systems of extracting square and cube roots, and could solve quadratic and certain types of indeterminate equations." Mahavira's most noteworthy contribution is his treatment of fractions for the first time and his rule for dividing one fraction by another, which did not appear in Europe until the 16th century.
B. B. Dutta writes: "The use of symbols-letters of the alphabet to denote unknowns, and equations are the foundations of the science of algebra. The Hindus were the first to make systematic use of the letters of the alphabet to denote unknowns. They were also the first to classify and make a detailed study of equations. Thus they may be said to have given birth to the modern science of algebra."

The great Indian mathematician Bhaskaracharya (1150 C.E.) produced extensive treatises on both plane and spherical trigonometry and algebra, and his works contain remarkable solutions of problems which were not discovered in Europe until the seventeenth and eighteenth centuries. He preceded Newton by over 500 years in the discovery of the principles of differential calculus.

For more refer to The Infinitesimal Calculus: How and Why it Was Imported into Europe - By C. K. Raju and
Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibh?s? - By C. K. Raju

A. L. Basham writes further, "The mathematical implications of zero (sunya) and infinity, never more than vaguely realized by classical authorities, were fully understood in medieval India. Earlier mathematicians had taught that X/0 = X, but Bhaskara proved the contrary. He also established mathematically what had been recognized in Indian theology at least a millennium earlier: that infinity, however divided, remains infinite, represented by the equation /X = ."

In the 14th century, Madhava, isolated in South India, developed a power series for the arc tangent function, apparently without the use of calculus, allowing the calculation of to any number of decimal places (since arc tan 1 = /4). Whether he accomplished this by inventing a system as good as calculus or without the aid of calculus; either way it is astonishing. Stanley Wolpert says: " An untutored Kerala mathematician named Madhava developed his own system of calculus, based on his knowledge of trigonometry around A.D. 1500, more than a century before either Newton or Liebnitz.

(source: An Introduction to India - By Stanley Wolpert p. 195).

By the fifteenth century C. E. use of the new mathematical concepts from India had spread all over Europe to Britain, France, Germany, and Italy, among others. A. L. Basham states also that

"The debt of the Western world to India in this respect [the field of mathematics] cannot be overestimated. Most of the great discoveries and inventions of which Europe is so proud would have been impossible without a developed system of mathematics, and this in turn would have been impossible if Europe had been shackled by the unwieldy system of Roman numerals. The unknown man who devised the new system was, from the world's point of view, after the Buddha, the most important son of India. His achievement, though easily taken for granted, was the work of an analytical mind of the first order, and he deserves much more honor than he has so far received."

Carl Friedrich Gauss ( 1777-1855), German scientist and mathematician, was considered as the "prince of mathematics. He is frequently called the founder of modern mathematics, who also studied Sanskrit.

Gauss "was said to have lamented that Archimedes in the third century B.C. E. had failed to foresee the Indian system of numeration; how much more advanced science would have been."

Unfortunately, Eurocentrism has effectively concealed from the common man the fact that we owe much in the way of mathematics to ancient India.

In ancient India, mathematics served as a bridge between understanding material reality and the spiritual conception. Vedic mathematics differs profoundly from Greek mathematics in that knowledge for its own sake (for its aesthetic satisfaction) did not appeal to the Indian mind. The mathematics of the Vedas lacks the cold, clear, geometric precision of the West; rather, it is cloaked in the poetic language which so distinguishes the East. Vedic mathematicians strongly felt that every discipline must have a purpose, and believed that the ultimate goal of life was to achieve self-realization and love of God and thereby be released from the cycle of birth and death.

After this period, India was repeatedly raided by muslims and other rulers and there was a lull in scientific research. Industrial revolution and Renaissance passed India by. Before Ramanujan, the only noteworthy mathematician was Sawai Jai Singh II, who founded the present city of Jaipur in 1727 A.D. This Hindu king was a great patron of mathematicians and astronomers. He is known for building observatories (Jantar Mantar) at Delhi, Jaipur, Ujjain, Varanasi and Mathura. Among the instruments he designed himself are Samrat Yantra, Ram Yantra and Jai Parkash.

More recently, intuitive Indian mathematical genius Srinivas Ramanujan (1887-1920), a friend to all numbers, was invited to Cambridge by Prof. G. H. Hardy, who recognized his brilliance at the sight of his first equation solution. Julian Huxley called Ramanujan "the greatest mathematician of the century." At the age of thirty he developed a formula for partitioning any natural number, which led to the solving of the Waring problem, expressing an integer as the sum of squares, cubes, or higher powers of a few integers. One day Hardy complained about the cab number that brought him to visit Ramanujan, "1729" as a dull number. Ramanujan responded instantly, " No Hardy, 1729 is a wonderful number! That is the only number which is the sum of two different sets of cubes, 1 and 12, and 9 and 10."

(source: An Introduction to India - By Stanley Wolpert p. 195).

Mountstuart Elphinstone wrote: "Their geometrical skill is shown among other forms by their demonstrations of various properties of triangles, especially one which expresses the area in the terms of the three sides, and was unknown in Europe till published by Clavius, and by their knowledge of the proportions of the radius to the circumference of a circle, which they express in a mode peculiar to themselves, by applying one measure and one unit to the radius and circumference. This proportion, which is confirmed by the most approved labors of Europeans, was not known out of India until modern times!"

(source: History of India - By Mountstuart Elphinstone London: John Murray Date of Publication: 1849 p. 130).

For more refer to The Infinitesimal Calculus: How and Why it Was Imported into Europe - By C. K. Raju and
Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibh?s? - By C. K. Raju

 

Srinivas Ramanujan: A Life of the Genius

Ramanujan is one of India?s great intellectual heroes, a mathematical genius who attributed his brilliance to a personal relationship with a Hindu Goddess - Namagiri. His work has been used to help unravel knots as varied as polymer chemistry and cancer, yet how he arrived at this theorems is still unknown. By age twelve he had mastered trigonometry so completely that he was inventing sophisticated theorems that astonished teachers. Mathematicians have mined his theorems ever since. They've figured out how to prove them. They've put them to use. Only recently, a lost bundle of his notebooks turned up in a Cambridge library. That set mathematics off on a whole new voyage of discovery. And where did all this unproven truth come from? Ramanujan was quick to tell us. He simply prayed to Sarasvathi, the Goddess of Learning, and she informed him.

His twenty-one major mathematical papers are still being plumbed for their secrets, and many of his ideas are used today in cosmology and computer science. The unsettling thing is, none of us can find any better way to explain the magnitude of his eerie brilliance.

(source: http://www.uh.edu/engines/epi495.htm ) John H. Lienhard (source: The Man Who Knew Infinity: A Life of the Genius Ramanujan - by Robert Kanigel)..(source: Ramanujan and Computing the Mathematical face of God). For more on Ramanuja, refer to chapter on Quotes321_340).

Vedic Mathematics

"Vedic Mathematics" is the name given to the ancient system of mathematics, or, to be precise, a unique technique of calculations based on simple rules and principles, with which any mathematical problem — be it arithmetic, algebra, geometry or trigonometry — can be solved. The system is based on 16 Vedic sutras or aphorisms, which are actually word-formulae describing natural ways of solving a whole range of mathematical problems. Some examples of sutras are "By one more than the one before", "All from 9 & the last from 10", and "Vertically & Crosswise". These 16 one-line formulae originally written in Sanskrit, which can be easily memorized, enables one to solve long mathematical problems quickly.

Born in the Vedic Age, but buried under centuries of debris, this remarkable system of calculation was deciphered towards the beginning of the 20th century, when there was a great interest in ancient Sanskrit texts, especially in Europe. However, certain texts called Ganita Sutras, which contained mathematical deductions, were ignored, because no one could find any mathematics in them. These texts, it's believed, bore the germs of what we now know as Vedic Mathematics.

Vedic math was rediscovered from the ancient Indian scriptures between 1911 and 1918 by Sri Bharati Krishna Tirthaji (1884-1960), a scholar of Sanskrit, Mathematics, History and Philosophy. He studied these ancient texts for years, and after careful investigation was able to reconstruct a series of mathematical formulae called sutras.

Bharati Krishna Tirthaji, who was also the former Shankaracharya of Puri, India, delved into the ancient Vedic texts and established the techniques of this system in his pioneering work — Vedic Mathematics (1965), which is considered the starting point for all work on Vedic math. It is said that after Bharati Krishna's original 16 volumes of work expounding the Vedic system were lost, in his final years he wrote this single volume, which was published five years after his death.

(source: Vedic Mathematics - about.com). For more refer to chapter on Glimpses VIII and Vedic Math websites).

 

The Indic Mathematical tradition

Jean-?tienne Montucla (1725-1799) French author of Histoire des mathematiques (1798):

“The ingenious number-system, which serves as the basis for modern arithmetic, was used by the Arabs long before it reached Europe . It would be a mistake, however, to believe that this invention is Arabic. There is a great deal of evidence, much of it provided by the Arabs themselves that this arithmetic originated in India .” [Montucla, I, p. 375J.

John Walls (1616-1703) referred to the nine numerals as Indian figures [Wallis (1695), p. 10]

Cataneo (1546) le noue figure de gli Indi, “the nine figures from India ”. [Smith and Karpinski (1911), p.3

Willichius (1540) talks of Zyphrae! Nice, “Indian figures”. [Smith and Karpinski (1911) p. 3]

The Crafte of Nombrynge (c. 1350), the oldest known English arithmetical tract: II fforthermore ye most vndirstonde that in this craft ben vsed teen figurys, as here bene writen for esampul 098 ^ 654321... in the quych we vse teen figwys of Inde. Questio II why Zen figurys of Inde? Soiucio. For as I have sayd afore thei werefondefrrst in Inde. [D. E. Smith (1909).

Petrus of Dada (1291) wrote a commentary on a work entitled Algorismus by Sacrobosco (John of Halifax, c. 1240), in which he says the following (which contains a mathematical error): Non enim omnis numerus per quascumquefiguras Indorum repraesentatur “Not every number can be represented in Indian figures”. [Curtze (1.897), p. 25.

Around the year 1252, Byzantine monk Maximus Planudes (1260—1310) composed a work entitled Logistike Indike (“Indian Arithmetic”) in Greek, or even Psephophoria kata Indos (“The Indian way of counting”), where he explains the following: “There are only nine figures.

These are: 123456789 - [figures given in their Eastern Arabic form]

A sign known as tziphra can be added to these, which, according to the Indians, means ‘nothing’. The nine figures themselves are Indian, and tziphra is written thus: 0”. [B. N., Pans. Ancien Fonds grec, Ms 2428, f” 186 r”]

Around 1240, Alexandre de Ville-Dieu composed a manual in verse on written calculation (algorism). Its title was Carmen de Algorismo, and it began with the following two lines: Haec algorismus ars praesens dicitur, in qua Talibus Indorumfruimur bis quinquefiguris.

“Algorism is the art by which at present we use those Indian figures, which number two times five”. [Smith and Karpinski (1911), p. 11]

In 1202, Leonard of Pisa (known as Fibonacci), after voyages that took him to the Near East and Northern Africa, and in particular to Bejaia (now in Algeria), wrote a tract on arithmetic entitled Liber Abaci (“a tract about the abacus”), in which he explains the following:

“Cum genitor meus a patria publicus scriba in duana bugee pro pisanis mercatoribus ad earn confluentibus preesset, me in pueritia mea ad se uenire faciens, inspecta utilitate el cornmoditate fiutura, ibi me studio abaci per aliquot dies stare uoluit et doceri. Vbi a mirabii magisterio in arte per nouem figuras Indorum introductus. . . Novem figurae Indorum hae sun!: cum his itaque novemfiguris. et turn hoc signo o. Quod arabice zephirum appellatur, scribitur qui libel numerus: “My father was a public scribe of Bejaia, where he worked for his country in Customs, defending the interests of Pisan merchants who made their fortune there. He made me learn how to use the abacus when I was still a child because he saw how I would benefit from this in later life. In this way I learned the art of counting using the nine Indian figures... The nine Indian figures are as follows:

987654321 - [figures given in contemporary European cursive form].

“That is why, with these nine numerals, and with this sign 0, called zephirum in Arab, one writes all the numbers one wishes.”[Boncompagni (1857), vol.1]

Rabbi Abraham Ben MeIr Ben Ezra (1092—1167), after a long voyage to the East and a period spent in Italy , wrote a work in Hebrew entitled: Sefer ha mispar (“Number Book”), where he explains the basic rules of written calculation. He uses the first nine letters of the Hebrew alphabet to represent the nine units. He represents zero by a little circle and gives it the Hebrew name of galgal (“wheel”), or, more frequently, sfra (“void”) from the corresponding Arabic word. However, all he did was adapt the Indian system to the first nine Hebrew letters (which he naturally had used since his childhood).

In the introduction, he provides some graphic variations of the figures, making it clear that they are of Indian origin, after having explained the place-value system: “That is how the learned men of India were able to represent any number using nine shapes which they fashioned themselves specifically to symbolize the nine units.” (Silberberg (1895), p.2: Smith and Ginsburg (1918): Steinschneider (1893).

Around the same time, John of Seville began his Liberalgoarismi de practica arismetrice (“Book of Algoarismi on practical arithmetic”) with the following:

“Numerus est unitatum cot/echo, quae qua in infinitum progredilur (multitudo enim crescit in infinitum), ideo a peritissimis Indis sub quibusdam regulis et certis lirnitibus infinita numerositas coarcatur, Ut de infinitis dfinita disciplina traderetur etfuga subtilium rerum sub alicuius artis certissima Jege ten eretur:

“A number is a collection of units, and because the collection is infinite (for multiplication can continue indefinitely), the Indians ingeniously enclosed this infinite multiplicity within certain rules and limits so that infinity could be scientifically defined: these strict rules enabled them to pin down this subtle concept.
[B. N., Paris, Ms. lat. 16 202, p 51: Boncompagni (1857), vol. I, p. 261

C. 1143, Robert of Chester wrote a work entitled: Algoritmi de numero Indorum (“Algoritmi: Indian figures”), which is simply a translation of an Arabic work about Indian arithmetic. [Karpinski (1915); Wallis (1685). p. 121

C. 1140, Bishop Raymond of Toledo gave his patronage to a work written by the converted Jew Juan de Luna and archdeacon Domingo Gondisalvo: the Liber Algorismi de numero Indorum (“Book of Algorismi of Indian figures) which is simply a translation into a Spanish and Latin version of an Arabic tract on Indian arithmetic. [Boncompagni (1857), vol. 11

C. 1130, Adelard of Bath wrote a work entitled: Algoritmi de numero Indorum (“Algoritmi: of Indian figures”), which is simply a translation of an Arabic tract about Indian calculation. [Boncompagni (1857), vol. Ii

C. 1125, The Benedictine chronicler William of Malmesbury wrote De gestis regum Anglorum, in which he related that the Arabs adopted the Indian figures and transported them to the countries they conquered, particularly Spain. He goes on to explain that the monk Gerbert of Aurillac, who was to become Pope Sylvester II (who died in 1003) and who was immortalized for restoring sciences in Europe, studied in either Seville or Cordoba , where he learned about Indian figures and their uses and later contributed to their circulation in the Christian countries of the West.

L Malmesbury (1596), f” 36 r’; Woepcke (1857), p. 35J

Written in 976 in the convent of Albelda (near the town of Logro?o , in the north of Spain ) by a monk named Vigila, the Coda Vigilanus contains the nine numerals in question, but not zero. The scribe clearly indicates in the text that the figures are of Indian origin:

“Item de figuels aritmetice. Scire debemus Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et geometrica et ceteris liberalibu.c disciplinis concedere. Et hoc manif?stum at in novem figuris, quibus quibus designant unum quenque gradum cuiu.slibetgradus. Quatrum hec sunt forma:

9 8 7 6 5 4 3 2 1.

“The same applies to arithmetical figures. It should be noted that the Indians have an extremely subtle intelligence, and when it comes to arithmetic, geometry and other such advanced disciplines, other ideas must make way for theirs. The best proof of this is the nine figures with which they represent each number no matter how high. This is how the figures look:

9 8 7 6 5 4 3 2 1

(source: The Indic Mathematical tradition - By Kosla Vepa).