Overview of Mathematics in India - India, China, and Japan 500 BCE-1700 CE - A brief course - The history of mathematics

The history of mathematics: A brief course (2013)

Part IV. India, China, and Japan 500 BCE-1700 CE

Chapter 19. Overview of Mathematics in India

From archaeological excavations at Mohenjo Daro and Harappa on the Indus River in Pakistan it is known that an early civilization existed in this region for about a millennium starting in 2500 BCE. This civilization may have been an amalgam of several different cultures, since anthropologists recognize five different physical types among the human remains. Many of the artifacts that were produced by this culture have been found in Mesopotamia, evidence of trade between the two civilizations. As a framework for the mathematical history we shall be studying in this chapter and the two following, we shall periodize this history as follows.

1. The Aryan Civilization. The early civilization of these five groups of people dis-appeared around 1500 BCE, and its existence was not known in the modern world until 1925. The cause of its extinction is believed to be an invasion from the northwest by a sixth group of people, who spoke a language closely akin to early Greek. Because of their language, these people are referred to as Aryans, a term that acquired a sinister racial meaning in the early twentieth century. (Used in this sense, it was a blemish on a popular brief history of mathematics by W. W. Rouse Ball.) The Aryans gradually expanded and formed a civilization of small kingdoms, which lasted about a millennium.

2. Sanskrit Literature. The language of the Aryans became a literary language known as Sanskrit, in which great classics of literature and science have been written. Sanskrit thus played a role in southern Asia analogous to that of Greek in the Mediterranean world and Chinese in much of eastern Asia.1 That is, it provided a means of communication among scholars whose native languages were not mutually comprehensible and a basis for a common literature in which cultural values could be preserved and transmitted. During the millennium of Aryan dominance, the spoken language of the people gradually diverged from written Sanskrit. Modern descendants of Sanskrit are Hindi, Gujarati, Bengali, and others. Sanskrit is the language of the Mahabharata and the Ramayana, two epic poems whose themes bear some resemblance to the Homeric epics, and of the Upanishads, which contain much of the moral teaching of Hinduism.

Among the most ancient works of literature in the world are the Hindu Vedas. The word means knowledge and is related to the English word wit. The composition of the Vedas began around 900 BCE, and additions continued to be made to them for several centuries. Some of these Vedas contain information about mathematics.

3. Hindu Religious Reformers. Near the end of the Aryan civilization, in the second half of the sixth century BCE, two figures of historical importance arise. One of these was Gautama Buddha (563–479), heir to a kingdom near the Himalaya Mountains, whose spiritual journey through life led to the principles of Buddhism. The other, named Mahavira (599–527), is less well known but has some importance for the history of mathematics. Like his contemporary Buddha, he began a reform movement within Hinduism. This movement, known as Jainism, still has several million adherents in India. It is based on a metaphysic that takes very seriously what is known in some Western ethical systems as the chain of being. Living creatures are ranked according to their awareness. Those having five senses are the highest, and those having only one sense are the lowest.

4. Islam in India. The rapid Muslim expansion from the Arabian desert in the seventh century brought Muslim invaders to India by the early eighth century. The southern valley of the Indus River became a province of the huge Umayyad Empire, but the rest of India preserved its independence, as it did 300 years later when another Muslim people, the Turks and Afghans, invaded. Still, the contact was enough to bring certain Hindu works, including the Hindu numerals, to the great center of Muslim culture in Baghdad. The complete and destructive conquest of India by the Muslims under Timur the Lame came at the end of the fourteenth century. Timur did not remain in India but sought new conquests; eventually he was defeated by the Ming dynasty in China. India was desolated by his attack and was conquered a century later by Akbar the Lion, the first of the Mogul emperors and a descendant of both Genghis Khan and Timur the Lame. The Mogul Empire lasted nearly three centuries and was a time of prosperity and cultural resurgence. One positive effect of this second Muslim expansion was a further exchange of knowledge between the Hindu and Muslim worlds. Interestingly, the official administrative language used for Muslim India was neither Arabic nor an Indian language; it was Persian.

5. British Rule. During the seventeenth and eighteenth centuries British and French trading companies were in competition for the lucrative trade with the Mogul Empire. British victories during the Seven Years War (1756–1763) left Britain in complete control of this trade. Coming at the time of Mogul decline due to internal strife among the Muslims and continued resistance on the part of the Hindus, this trade opened the door for the British to make India part of their empire. British colonial rule lasted nearly 200 years, coming to an end only after World War II. British rule made it possible for European scholars to become acquainted with Hindu classics of literature and science. Many Sanskrit works were translated into English in the early nineteenth century and became part of the world's science and literature.

6. Independent India. Some 65 years have now passed since India became an independent nation. This period has been one of great cultural and economic resurgence in India, and mathematics has benefited fully from this resurgence.

Within this general framework, we can distinguish three periods in the development of mathematics in the Indian subcontinent. The first period begins around 900 BCE with individual mathematical results forming part of the Vedas. The second begins with treatises called siddhantas, concerned mostly with astronomy but containing explanations of mathematical results, which appear in the second century CE. These treatises led to continuous progress for 1500 years, during which time much of algebra, trigonometry, and certain infinite series that now form part of calculus were discovered, a century or more before Europeans developed calculus. In the third stage, which began during the two centuries of British rule, this Hindu mathematics came to be known in the West, and Indian mathematicians began to work and write in the modern style of mathematics that is now universal. In the present chapter, we shall discuss this mathematical development in general terms, concentrating on a few of the major works and authors and their motivation, with mathematical details to follow in the succeeding chapters.

19.1 The Sulva Sutras

In the period from 800 to 500 BCE a set of verses of geometric and arithmetic content were written and became part of the Vedas. These verses are known as the Sulva Sutras or Sulba Sutras.2 The name means Cord Rules and probably reflects the use of a stretched rope or cord as a way of measuring length, as in Egypt. The root sulv originally meant to measure or to rule, although it also has the meaning of a cord or rope; sutra means thread or cord, a common measuring instrument. In the case of the Vedas the objects being measured with the cords were altars. The maintenance of altar fires was a duty for pious Hindus; and because Hinduism is polytheistic, it was necessary to consider how elaborate and large the fire dedicated to each deity was to be. This religious problem led to some interesting problems in arithmetic and geometry.

Two scholars who studied primarily the Sanskrit language and literature made important contributions to mathematics. Pingala, who lived around 200 BCE, wrote a treatise known as the Chandahsutra, containing one very important mathematical result, which, however, was stated so cryptically that one must rely on a commentary written 1200 years later to know what it meant. Later, a fifth-century scholar named Panini standardized the Sanskrit language, burdening it with some 4000 grammatical rules that make it many times more difficult to learn than any other Indo-European language. In the course of doing so, he made extensive use of combinatorics and the kind of abstract reasoning that we associate with algebra. These subjects set the most ancient Hindu mathematics apart from that of other nations.

19.2 Buddhist and Jain Mathematics

As with any religion that encourages quiet contemplation and the renunciation of sensual pleasure, Jainism often leads its followers to study mathematics, which provides a different kind of pleasure, one appealing to the mind. There have always been some mathematicians among the followers of Jainism, right down to modern times, including one in the ninth century bearing the same name as the founder of Jainism. This other Mahavira speculated on arithmetic operations that yield infinite or infinitesimal results, a topic of interest to Jains in connection with cosmology and physics (Plofker 2009, pp. 58, 163). The early work of Jain mathematicians is notable for algebra (the Sthananga Sutra, from the second century BCE), for its concentration on topics that are unique to early Hindu mathematics, such as combinatorics (the Bhagabati Sutra, from around 300 BCE), and for speculation on infinite numbers (the Anuyoga Dwara Sutra, probably from the first century BCE). The Jains were the first to use the square root of 10 as an approximation to the ratio of a circle to its diameter, that is, the number we call π (Plofker 2009, p. 59). Like the Jains, Buddhist monks were very fond of large numbers, and their influence was felt when Buddhism spread to China in the sixth century CE.

19.3 The Bakshali Manuscript

A birchbark manuscript unearthed in 1881 in the village of Bakshali, near Peshawar, Pakistan is believed by some scholars to date from the seventh century CE, although Sarkor (1982) believes it cannot be later than the end of the third century, since it refers to coins named dimgnimgra and dramma, which are undoubtedly references to the Greek coins known as the denarius and the drachma, introduced into India by Alexander the Great. These coins had disappeared from use in India by the end of the third century. Plofker, however (2009, p. 157) places it somewhere in the period 700–1200. The Bakshali manuscript contains some interesting algebra that will be discussed in Chapter 20.

19.4 The Siddhantas

During the second, third, and fourth centuries CE, Hindu scientists compiled treatises on astronomy known as siddhantas. The word siddhanta means a system.3 One of these treatises, the Surya Siddhanta (System of the Sun), from the late fourth century, has survived intact. Another from approximately the same time, the Paulisha Siddhanta, was frequently referred to by the Muslim scholar al-Biruni (973–1048). The name of this treatise seems to have been bestowed by al-Biruni, who says that the treatise was written by an Alexandrian astrologer named Paul.

19.5 Hindu–Arabic Numerals

The decimal system of numeration, in which 10 symbols are used and the value of a symbol depends on its physical location relative to the other symbols in the representation of a number, came to the modern world from India by way of the medieval Muslim civilization. These symbols have undergone some changes in their migration from ancient India to the modern world, as shown in the photo. The idea of using a symbol for an empty place was the final capstone on the creation of a system of counting and calculation that is in all essential aspects the one still in use. This step must have been taken well over 1500 years ago in India. There is some evidence, not conclusive, that symbols for an empty place were used earlier, but no such symbol occurs in the work of Arbyabhata I. On the other hand, such a symbol, called in Sanskrit sunya (empty), occurs in the work of Brahmagupta a century after Arybhata.

Evolution of theHindu–Arabic numerals from India to modern Europe. Copyright© Vandenhoeck & Ruprecht, from the book byKarl Menninger, Zahlwort und Ziffer, 3rd ed., Göttingen,1979.

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19.6 Aryabhata I

With the writing of treatises on mathematics and astronomy, we at last come to some records of the motives that led people to create Hindu mathematics, or at least to write expositions of it. A mathematician named Aryabhata (476–550), the first of two mathematicians bearing that name, lived in the late fifth and early sixth centuries at Kusumapura (now Pataliputra, a village near the city of Patna) and wrote a book called the Aryabhatiya. This work had been lost for centuries when it was recovered by the Indian scholar Bhau Daji (1822–1874) in 1864. Scholars had known of its existence through the writings of commentators and had been looking for it. Writing in 1817, the English scholar Henry Thomas Colebrooke (1765–1837), who translated other Sanskrit mathematical works into English, reported, “A long and diligent research of various parts of India has, however, failed of recovering any part of the. . .Algebra and other works of Aryabhata.” Ten years after its discovery the Aryabhatiya was published at Leyden and attracted the interest of European and American scholars. It consists of 123 stanzas of verse, divided into four sections, of which the first, third, and fourth are concerned with astronomy and the measurement of time.

Like all mathematicians, Aryabhata I was motivated by intellectual interest. This interest, however, was closely connected with his Hindu piety. He begins the Aryabhatiya with the following tribute to the Hindu deity:

Having paid reverence to Brahman, who is one but many, the true deity, the Supreme Spirit, Aryabhata sets forth three things: mathematics, the reckoning of time, and the sphere. [Clark, 1930, p. 1]

The translator adds phrases to explain that Brahman is one as the sole creator of the universe, but is many via a multitude of manifestations.

Aryabhata then continues his introduction with a list of the astronomical observations that he will be accounting for and concludes with a promise of the reward awaiting the one who learns what he has to teach:

Whoever knows this Dasagitika Sutra which describes the movements of the earth and the planets in the sphere of the asterisms passes through the paths of the planets and asterisms and goes to the higher Brahman. [Clark, 1930, p. 20]

As one can see, students in Aryabhata's culture had an extra reason to study mathematics and astronomy, beyond the concerns of practical life and the pleasures of intellectual edification. Learning mathematics and astronomy helped to advance the soul through the cycle of reincarnations that Hindus believed in.

After setting out his teaching on the three subjects, Aryabhata concludes with a final word of praise for the Hindu deity and invokes divine endorsement of his labors:

By the grace of God the precious sunken jewel of true knowledge has been rescued by me, by means of the boat of my own knowledge, from the ocean which consists of true and false knowledge. He who disparages this universally true science of astronomy, which formerly was revealed by Svayambhu4 and is now described by me in this Aryabhatiya, loses his good deeds and his long life. [Clark, 1930, p. 81]

19.7 Brahmagupta

The establishment of research centers for astronomy and mathematics at Kusumapura and Ujjain, near the geographical center of modern India, produced a succession of good mathematicians and mathematical works for many centuries after Aryabhata I. Half a century after the death of Aryabhata I, another Hindu mathematician, Brahmagupta (598–670), was born in the city of Sind, now in Pakistan. He was primarily an astronomer, but his astronomical treatise, the Brahmasphutasiddhanta (literally The Corrected Brahma Siddhanta), contains several chapters on computation (ganita). The Hindu interest in astronomy and mathematics continued unbroken for several centuries, producing important work on trigonometry in the tenth century.

19.8 Bhaskara II

Approximately 500 years after Brahmagupta, in the twelfth century, the mathematician Bhaskara (1114–1185), the second of that name, was born on the site of the modern city of Bijapur, in southwestern India. He is the author of the Siddhanta Siromani, in four parts, a treatise on algebra and geometric astronomy. Only the first of these parts, known as the Lilavati, and the second, known as the Vija Ganita,5 concern us here. Bhaskara says that his work is a compendium of knowledge, a sort of textbook of astronomy and mathematics. The name Lilavati was common among Hindu women. Many of the problems are written in the form of puzzles addressed to this Lilavati.

Bhaskara II apparently wrote the Lilavati as a textbook to form part of what we would call a liberal education. His introduction reads as follows:

Having bowed to the deity, whose head is like an elephant's [Ganesh], whose feet are adored by gods; who, when called to mind, relieves his votaries from embarrassment; and bestows happiness on his worshippers; I propound this easy process of computation, delightful by its elegance, perspicuous with words concise, soft and correct, and pleasing to the learned. [Colebrooke, 1817, p. 1]

As a final advertisement at the end of his book, Bhaskara extols the pleasure to be derived from learning its contents:

Joy and happiness is indeed ever increasing in this world for those who have Lilavati clasped to their throats, decorated as the members are with neat reduction of fractions, multiplication, and involution, pure and perfect as are the solutions, and tasteful as is the speech which is exemplified. [Colebrooke, 1817, p. 127]

The Vija Ganita consists of nine chapters, in the last of which Bhaskara tells something about himself and his motivation for writing the book:

On earth was one named Maheswara, who followed the eminent path of a holy teacher among the learned. His son Bhaskara, having from him derived the bud of knowledge, has composed this brief treatise of elemental computation. As the treatises of algebra [vija ganita] by Brahmagupta, Shidhara and Padmanabha are too diffusive, he has compressed the substance of them in a well-reasoned compendium for the gratification of learners. . .to augment wisdom and strengthen confidence. Read, do read, mathematician, this abridgement, elegant in style, easily understood by youth, comprising the whole essence of computation, and containing the demonstration of its principles, replete with excellence and void of defect. [Colebrooke, 1817, pp. 275–276]

The mathematician “Shidhara” is probably Sridhara (870–930). Information on a mathematician named Padmanabha does not appear to be available.

19.9 Muslim India

Indian mathematical culture reflects the religious division between the Muslim and Hindu communities to some extent. The Muslim conquest brought Arabic and Persian books on mathematics to India. Some of these works were translated from ancient Greek, and among them was Euclid's Elements. These translations of later editions of Euclid contained certain obscurities and became the subject of commentaries by Indian scholars. Akbar the Lion decreed a school curriculum for Muslims that included three-fourths of what was known in the West as the quadrivium. Akbar's curriculum included arithmetic, geometry, and astronomy, leaving out only music.6 Details of this Indian Euclidean tradition are given in the paper by De Young (1995).

19.10 Indian Mathematics in the Colonial Period and After

One of the first effects of British rule in India was to acquaint European scholars with the treasures of Hindu mathematics described above. A century passed before the British colonial rulers began to establish European-style universities in India. According to Varadarajan (1983), these universities were aimed at producing government officials, not scholars. As a result, one of the greatest mathematical geniuses of all time, Srinivasa Ramanujan (1887–1920), was not appreciated and had to appeal to mathematicians in Britain to gain a position that would allow him to develop his talent. The necessary conditions for producing great mathematics were present in abundance, however, and the establishment of the Tata Institute in Bombay (now Mumbai) and the Indian Statistical Institute in Calcutta were important steps in this direction. After Indian independence was achieved, the first prime minister, Jawaharlal Nehru (1889–1964), made it a goal to achieve prominence in science. This effort has been successful in many areas, including mathematics. The names of Komaravolu Chandrasekharan (b. 1920), Harish-Chandra (1923–1983), and others have become celebrated the world over for their contributions to widely diverse areas of mathematics.

19.10.1 Srinivasa Ramanujan

The topic of power series is one in which Indian mathematicians had anticipated some of the discoveries in seventeenth-and eighteenth-century Europe. It was a facility with this technique that distinguished Ramanujan, who taught himself mathematics after having been refused admission to universities in India. After publishing a few papers, starting in 1911, he was able to obtain a stipend to study at the University of Madras. In 1913 he took the bold step of communicating some of his results to G. H. Hardy (1877–1947). Hardy was so impressed by Ramanujan's ability that he arranged for Ramanujan to come to England. Thus began a collaboration that resulted in seven joint papers with Hardy, while Ramanujan alone was the author of some 30 others. He rediscovered many important formulas and made many conjectures about functions such as the hypergeometric function that are represented by power series.

Unfortunately, Ramanujan was in frail health, and the English climate did not agree with him. Nor was it easy for him to maintain his devout Hindu practices so far from his normal Indian diet. He returned to India in 1919, but succumbed to illness the following year. Ramanujan's notebooks have been a subject of continuing interest to mathematicians. Hardy passed them on to G. N. Watson (1886–1965), who published a number of “theorems stated by Ramanujan.” The full set of notebooks was published in the mid-1980s (see Berndt, 1985).

Questions

Historical Questions

19.1. What application motivates the mathematics included in the Sulva Sutras?

19.2. What mathematical subjects studied by Indian mathematicians long ago have no counterpart in the other cultures studied up to this point?

19.3. Which physical science is most closely connected with mathematics in the Hindu documents?

19.4. What justifications for the study of mathematics do the Hindu authors Aryabhata I and Bhaskara II mention?

Questions for Reflection

19.5. What differences do you notice in the “style” of mathematics in Greece and India? Consider in particular the importance of logic, the metaphysical views of the nature of such things as lines, circles, and the like, and the interpretation of the infinite.

19.6. One reflection of Mesopotamian influence in India is the division of the circle into 360 degrees. Does having this system in common indicate that the Hindus received their knowledge of trigonometry from the Greeks?

19.7. Archimedes wrote a work called the Sand-reckoner to prove that the universe (as the Greeks pictured it) could be filled with a finite number of grains of sand. The necessity of doing so shows that the Greeks had the same psychological difficulties that all people have in distinguishing clearly between “infinite” and “very large.” In the following passage from a Jain work, a related issue is addressed, namely what is the largest nameable number?

Consider a trough whose diameter is of the size of the earth. Fill it up with white mustard seeds counting them one after another. Similarly, fill up with mustard seeds other troughs of the sizes of the various lands and seas. Still it is difficult to reach the highest enumerable number.

Should the infinite be thought of as in some sense “approximated” by a very large finite quantity, or is it qualitatively different? Is it possible to create a meaningful arithmetic in which there is a largest integer?

19.8. Are there any clues in the cultural context of Indian mathematics that help to explain why it was the only ancient civilization to develop a system of numeration that was based on both the number 10 and place value, so that only 10 symbols were needed to write it?

Notes

1. India also exerted a huge cultural influence on southeast Asia, through the Buddhist and Hindu religions and in architecture and science. Moreover, both cultural contact and commercial contact between India and China have a long history.

2. The Sulva Sutras are discussed in many places. The reader is cautioned against books discussing “Vedic mathematics,” however—for example, Maharaja (1965), which presents elaborate modern mathematical arguments tenuously connected to the Vedas and alleges that analytic geometry in its modern form, which associates an equation with a curve, was known to the Vedic authors 2500 years ago. Communication problems occur even in generally reliable sources, such as the book of Srinivasiengar (1967), which is the source of many of the facts reported in this chapter and the next. (Everything in these two chapters comes from some secondary source, usually the books of Srinivasiengar, Plofker, Colebrooke, or Clark.) Srinivasiengar asserts (p. 6) that the unit of length known as the vyayam was “about 96 inches,” and “possibly this represented the height of the average man in those days.” This highly improbable statement results from the imprecision in the term height. According to Plofker (2009, p. 18), there was a unit called man height, but it meant the height a man could reach into the air standing on the ground. Even with that clarification, 96 inches seems improbable.

3. A colleague of the author suggested that this word may be cognate with the Greek idimgn (neuter plural idónta), the aorist participle of the verb meaning see, which can be translated as “after seeing. . .”.

4. According to the Matsya Purana, the sixteenth purana of the Hindu scriptures, Svayambhu was a self-generated deity who infused the universe with the potential to generate life.

5. This Sanskrit word means literally seed computation, the word seed being used in the algebraic sense of root. It is compounded from the Sanskrit root vij- or bij-, which means seed. As we have stated many times, the basic idea of algebra is to name explicitly one or more numbers (the “seed”) given certain implicit descriptions of them (metaphorically, “flowers” that they produce), usually the result of operating on them in various ways. The word is usually translated as algebra.

6. The quadrivium is said to have been proposed by Archytas, who apparently communicated it to Plato when the latter was in Sicily to consult with the ruler of Syracuse; Plato incorporated it in his writings on education, as discussed in Chapter 12.