Indian mathematics

Indian mathematics emerged in the Indian subcontinent until the end of the 18th century. In the classical period of Indian mathematics (400 AD to 1200 AD), important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II.

B

 * The Indian system of counting is probably the most successful intellectual innovation ever devised by human beings. It has been universally adopted. ...It is the nearest thing we have to a universal language.
 * John D. Barrow, The Book of Nothing (2009) chapter one "Zero—The Whole Story"


 * Medieval Indian mathematicians, such as Brahmagupta (7th century), Mahavira (9th century) and Bhüskara (19th 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.
 * A. L. Basham,  in The Wonder That Was India


 * Through the necessity of accurately laying out the open-air site of a sacrifice Indians very early evolved a simple system of geometry, but in the sphere of practical knowledge the-world owes most to India in the realm of mathematics, which were developed in Gupta times to a stage more advanced than that reached by any other nation of antiquity. ‘The success of Indian mathematics was mainly due to the fact that the Indians had a clear conception of abstract number, as distinct from numerical quantity of objects or spatial extension. While Greek mathematical science was largely based on mensuration and geometry, India transcended these conceptions quite early, and, with the aid of a simple numeral notation, devised a rudimentary algebra which allowed more complicated calculations than were  possible to the Greeks, and led to the study of number for its own sake.
 * A. L. Basham,  in The Wonder That Was India


 * "The cord stretched in the diagonal of an oblong produces both [areas] which the cords forming the longer and shorter sides of an oblong produce separately"
 * (Baudhayana sutra 1.48; Apastambd sutra 1.4; Katyayana sutra 2.11). References from Seidenberg 1983, 98. ** in Bryant, E. F. (2001). The Quest for the Origins of Vedic Culture : the Indo-Aryan migration debate. Oxford University Press. chapter 12

D

 * 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.
 * Will Durant, Our Oriental heritage

E

 * 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.
 * Sir Mountstuart Elphinstone, attributed at, ( Sanskrit Civilization - By G. R. Josyer p. 2).


 * We owe a lot to the Indians who taught us how to count, without which no worthwhile scientific discovery could have been made.
 * Albert Einstein, source: Ignited Minds: Unleashing the Power Within India, A.P.J. Abdul Kalam. Quoted from Gewali, Salil (2013). Great Minds on India. New Delhi: Penguin Random House.


 * In order to instill a proper and well-founded pride in Hindus, it is (once more) most important to restore the truth about Hindu history, especially about Hindu society's glorious achievements. In technology, it cannot match China, which was the world leader until a mere three, four centuries ago. But in abstract sciences like linguistics, logic, mathematics, Hindu culture has been the chief pioneer.
 * Elst, Koenraad (1991). Ayodhya and after: Issues before Hindu society.


 * In the Vedic Age, India was very religious, but it was also ahead of the rest in mathematics and astronomy. Thus, the geometry of the Shulba Sutras, 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. Modern Hindus are fond of recalling this scientific element in their tradition, e.g. by quoting Carl Sagan: “Hindu cosmology gives a time-scale for the earth and the universe which is consonant with that of modern scientific cosmology”, as opposed to the limited Biblical-Quranic cosmology, which was protected against more far-sighted alternatives by a vigilant religious orthodoxy.
 * Decolonizing the Hindu Mind, 2001, p. 29-30, by Koenraad Elst


 * In the Shulba Sutra appended to Baudhayana’s Shrauta Sutra, mathematical instructions are given for the construction of Vedic altars. One of its remarkable contributions is the theorem usually ascribed to Pythagoras, first for the special case of a square (the form in which it was discovered), then for the general case of the rectangle: “The diagonal of the rectangle produces the combined surface which the length and the breadth produce separately.”
 * Elst, Koenraad (1999). Update on the Aryan invasion debate New Delhi: Aditya Prakashan.

H

 * In the whole history of Mathematics, there has been no more revolutionary step than the one which the Indian made when they invented the sign ‘0’ for the empty column of the counting frame.
 * Lancelot Thomas Hogben. source: Mathematics for the Million, Lancelot Thomas Hogben. Quoted from Gewali, Salil (2013). Great Minds on India. New Delhi: Penguin Random House.


 * Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development... 1. ...Rhetoric Algebra, or "reckoning by complete words." ...the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming... continuous prose. ...2. ...Syncopated Algebra... is essentially rhetorical and therein like the first in its treatment of questions, but we now find for often-recurring operations and quantities certain abbreviational symbols. ...3. ...Symbolic Algebra ...uses a complete system of notation by signs having no visible connection with the words or things which they represent, a complete language of symbols, which supplants entirely the rhetorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a conjunction here and there, and so on. Neither is it the Europeans posterior to the middle of the seventeenth century who were the first to use Symbolic forms of Algebra. In this they were anticipated many centuries by the Indians.
 * Sir Thomas Little Heath, Diophantos of Alexandria: A Study in the History of Greek Algebra (1885) pp. 78-79.

I

 * 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.
 * Georges Ifrah. source: The Universal History of Numbers, Georges Ifrah. Quoted from Gewali, Salil (2013). Great Minds on India. New Delhi: Penguin Random House.


 * Ancient Indian culture has regarded the science of numbers as the noblest of its arts … A thousand years ahead of Europeans, Indian savants knew that zero and infinity were mutually inverse notions. In short, Indian science was born out of a mystical and religious culture and the etymology of the Sanskrit word used to describe numbers and the science of numbers bears witn 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.
 * Georges Ifrah. source: The Universal History of Numbers, Georges Ifrah. Quoted from Gewali, Salil (2013). Great Minds on India. New Delhi: Penguin Random House.

L

 * ''C'est de l'Inde que nous vient l'ingénieuse méthode d'exprimer tous les nombres avec dix caractères, en leur donnant à la fois, une valeur absolue et une valeur de position; idée fine et importante, qui nous paraît maîntenant si simple, que nous en sentons à peine, le mérite. Mais cette simplicité même, et l'extrême facilité qui en résulte pour tous les calculs, placent notre système d'arithmétique au premier rang des inventions utiles; et l'on appréciera la difficulté d'y parvenir, si l'on considère qu'il a échappé au génie d'Archimède et d'Apollonius, deux des plus grands hommes dont l'antiquité s'honore.
 * It is India that gave us the ingenious method of expressing all numbers using ten characters, giving these numbers simultaneously a value absolute and a value of position; a fine and important idea, which seems so simple now, that we hardly appreciate its merit. But this very simplicity, the extreme ease resulting in all calculations, place our system of arithmetic in the first rank of useful inventions; and we appreciate the difficulty of achieving this, considering that it escaped the genius of Archimedes and Apollonius, two of the greatest and most honored men of antiquity.''
 * Pierre-Simon Laplace, Exposition du Système du Monde, Vol. 2 (1798) also quoted in Tobias Dantzig, Number: The Language of Science (1930).

O

 * My confidence in our shared future is grounded in my respect for India’s treasured past—a civilization that has been shaping the world for thousands of years. Indians unlocked the intricacies of the human body and the vastness of our universe. And it is no exaggeration to say that our information age is rooted in Indian innovations—including the number zero.
 * Barack Obama. Quoted from Gewali, Salil (2013). Great Minds on India. New Delhi: Penguin Random House.

M

 * To the Hindus is due the invention of algebra and geometry, and their application to astronomy.
 * Sir Monier-Williams attributed, in (source: Indian Wisdom - By Monier Williams p. 185).


 * I once heard, and I think it is true, that only one man in the world—some Indian mathematician—understood the mathematics of string theory in eleven-dimensional space, and he dreamed it.
 * Mullis, Kary B - Dancing Naked in the Mind Field-Knopf Doubleday Publishing Group_Vintage Books (2010)

N

 * We know that the trigonometric sine is not mentioned by Greek mathematicians and astronomers, that it was used in India from the Gupta period onwards (third century).... The only conclusion possible is that the use of sines is an Indian development and not a Greek one. But Tannery, persuaded that the Indians could not have made any mathematical inventions, preferred to assume that the sine was a Greek idea not adopted by Hipparchus, who gave only a cable of chords. For Tannery, the fact that the Indians knew of sines was sufficient proof that they must have heard about them from the Greeks.
 * Joseph Needham quoted in   R.C. Majumdar, ‘Nationalist Historians’, in Philips, ed., Historians of India, Pakistan and Ceylon. quoted from E. Sreedharan - A Textbook of Historiography, 500 B.C. to A.D. 2000.


 * 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.
 * Professor H. G. Rawlinson writes: (Legacy of India 1937, p. 5). attributed at

S

 * However, Seidenberg was told by the indologists that these Sutras, or any Vedic text for that matter, were definitely written later than 1700 BC. But mathematical data cannot be manipulated just like that, and Seidenberg remained convinced of his case: “Whatever the difficulty there may be [concerning chronology], it is small in comparison with the difficulty of deriving the Vedic ritual application of the theorem from Babylonia. (The reverse derivation is easy)… the application involves geometric algebra, and there is no evidence of geometric algebra from Babylonia.  And the geometry of Babylonia is already secondary whereas in India it is primary.” [To satisfy the indologists, he said that the Shulba Sutra had conserved an older tradition, and that it is from this one that the Babylonians had learned their mathematics:] “Hence we do not hesitate to place the Vedic (…) rituals, or more exactly, rituals exactly like them, far back of 1700 BC. (…) elements of geometry found in Egypt and Babylonia stem from a ritual system of the kind described in the Sulvasutras.”
 * A. Seidenberg: “The ritual origin of geometry”, Archive for History of Exact Sciences, 1962, p. 488-527, specifically p-515, quoted by N.S. Rajaram and D. Frawley: Vedic Aryans’ and the Origins of Civilization, WH Press, Québec 1995, p-85. Seidenberg: “The ritual origin of geometry”, Archive for History of Exact Scieces, 1962, p.515, quoted by N.S. Rajaram and D. Frawley: Vedic ‘Aryans’ and the Origins of Civilization, p.85., quoted in Elst, Koenraad (1999). Update on the Aryan invasion debate New Delhi: Aditya Prakashan.


 * There is not, and cannot be, number as such. There are several number worlds as there are several Cultures. We find an Indian, an Arabian, a Classical, a Western type of mathematical thought and, corresponding with each, a type of number — each type fundamentally peculiar and unique, an expression of a specific world feeling, a symbol having a specific validity which is even capable of scientific definition, a principle of ordering the Become which reflects the central essence of one and only one soul, viz., the soul of that particular Culture. Consequently, there are more mathematics than one. ... and so it is understandable that even negative numbers, which to us offer no conceptual difficulty, were impossible in the Classical mathematic, let alone zero as a number, that refined creation of a wonderful abstractive power which, for the Indian soul that conceived it as base for a positional numeration, was nothing more nor less than the key to the meaning of existence. Negative magnitudes have no existence.... But when we are told that probably (it is at best a doubtful venture to meditate upon so alien an expression of Being) the Indians conceived numbers which according to our ideas possessed neither value nor magnitude nor relativity, and which only became positive and negative, great or small units in virtue of position, we have to admit that it is impossible for us exactly to re-experience what spiritually underlies this kind of number. For us, 3 is always something, be it positive or negative; for the Greeks it was unconditionally a positive magnitude, +3; but for the Indian it indicates a possibility without existence, to which the word “something” is not yet applicable, outside both existence and non-existence which are properties to be introduced into it. +3, -3, 1/3, are thus emanating actualities of subordinate rank which reside in the mysterious substance (3) in some way that is entirely hidden from us. It takes a Brahmanic soul to perceive these numbers as self-evident, as ideal emblems of a self-complete world form; to us they are as unintelligible as is the Brahman Nirvana, for which, as lying beyond life and death, sleep and waking, passion, compassion and dispassion and yet somehow actual, words entirely fail us. Only this spirituality could originate the grand conception of nothingness as a true number, zero, and even then this zero is the Indian zero for which existent and non-existent are equally external designations.
 * Oswald Spengler, Decline of the West


 * I shall not now speak of the knowledge of the Hindus … of their suitable discoveries in the science of astronomy—discoveries even more ingenious than those of the Greeks and Babylonians, of their rational system of mathematics, or of their method of calculation which no words can praise strongly enough; I mean the system using nine symbols.
 * Severus Sebokht, quoted in The Wonder That Was India, A.L. Basham. Quoted from Gewali, Salil (2013). Great Minds on India. New Delhi: Penguin Random House.


 * I will omit all discussion of the science of the Hindus, a people not the same as the Syrians, their subtle discoveries in the science of astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians; their computing that surpasses description. I wish only to say that this computation is done by means of nine signs. If those who believe because they speak Greek, that they have reaced the limits of science should know these things, they would be convinced that there are also others who know something.
 * Severus Sebokht,

V

 * Like the crests on the heads of peacocks, like the gems on the hoods of the cobras, mathematics is at the top of the Vedanga sastras.
 * Vedanga Jyotisa, 4