Amazing Science - 2

Amazing Science
The Corrosion Resistant Iron Pillar of Delhi
The pillar—over seven metres high and weighing more than
six tonnes—was erected by Kumara Gupta of Gupta
dynasty that ruled northern India in AD 320-540.

Press Trust of India
The Indian Express 26 January 2004

Experts at the Indian Institute of Technology have resolved the mystery behind the 1,600-year-old iron pillar in Delhi, which has never corroded despite the capital's harsh Metallurgists at Kanpur IIT have discovered that a thin layer of "misawite", a compound of iron, oxygen and hydrogen, has protected the cast iron pillar from rust.
The protective film took form within three years after erection of the pillar and has been growing ever so slowly since then. After 1,600 years, the film has grown just one-twentieth of a millimeter thick, according to
R. Balasubramaniam of the IIT.

In a report published in the journal Current Science Balasubramanian says, the protective film was formed catalytically by the presence of high amounts of phosphorous in the iron—as much as one per cent against less than 0.05 per cent in today's iron.
The high phosphorous content is a result of the unique iron-making process practiced by ancient Indians, who reduced iron ore into steel in one step by mixing it with charcoal.
Modern blast furnaces, on the other hand, use limestone in place of charcoal yielding molten slag and pig iron that is later converted into steel. In the modern process most phosphorous is carried away by the slag.
The pillar—over seven metres high and weighing more than six tonnes—was erected by Kumara Gupta of Gupta dynasty that ruled northern India in AD 320-540.
Stating that the pillar is "a living testimony to the skill of metallurgists of ancient India", Balasubramaniam said the "kinetic scheme" that his group developed for predicting growth of the protective film may be useful for modeling long-term corrosion behaviour of containers for nuclear storage applications
The Delhi iron pillar is testimony to the high level of skill achieved by ancient Indian iron smiths in the extraction and processing of iron. The iron pillar at Delhi has attracted the attention of archaeologists and corrosion technologists as it has withstood corrosion for the last 1600 years.

Minerals and Metals in Kautilya's Arthasastra
It is interesting to note that Kautilya prescribes that the state should carry out most of the businesses, including mining. No private enterprise for Kautilya! One is amazed at the breadth of Kautilya's knowledge. Though primarily it is treatise on statecraft, it gives detailed descriptions and instructions on geology, agriculture, animal husbandry, metrology etc. Its encyclopedic in its coverage and indicates that all these sciences were quite developed and systematized in India even 2500 years ago. It is surprising that even in the I Millennium BC, they had developed an elaborate terminology for different metals, minerals and alloys. Brass (arakuta) was known, so also steel (vrattu), bronze (kamsa), bell-metal (tala) was an alloy of copper with arsenic, but tin-copper alloy was known as trapu. A bewildering variety of jewellery was also classified and given distinctive names.
The chapter begins with the importance of 'mines and metals' in the society and here we are told that one of the most crucial statements in the Arthasastra is that gold, silver, diamonds, gems, pearls, corals, conch-shells, metals, salt and ores derived from the earth, rocks and liquids were recognized as materials coming under the purview of mines. The metallic ores had to be sent to the respective Metal Works for producing 'twelve kinds of metals and commodities'. Though the authors wish to show the importance of mines and metals in the society, yet what they point to is their importance for the state and the powers that the state exercised over them. Perhaps, Kautilya himself did not treat the matter so and focused to show its importance for the state alone as the book Arthasastra is on statecraft and not on society.
The next section deals with the gem minerals and is treated more extensively than others. We wonder if it is not due to the fact that the gem minerals reflected the richness of Indian kings. Here we are told that Mani-dhatu or the gem minerals were characterized in the Arthasastra as 'clear, smooth, lustrous, and possessed of sound, cold, hard and of a light color'. Excellent pearl gems had to be big, round, without a flat surface, lustrous, white, heavy, and smooth and perforated at the proper place. There were specific terms for different types of jewellery: Sirsaka (for the head, with one pearl in the centre, the rest small and uniform in size), avaghataka (a big pearl in the center with pearls gradually decreasing in size on both sides), indracchanda (necklace of 1008 pearls), manavaka (20 pearl string), ratnavali (variegated with gold and gems), apavartaka (with gold, gems and pearls at intervals), etc. Diamond (vajra) was discovered in India in the pre-Christian era. The Arthasastra described certain types of generic names of minerals red saugandhika, green vaidurya, blue indranila and colorless sphatika. Deep red spinel or spinel ruby identified with saugandhika, actually belongs to a different (spinel) family of minerals. Many other classes of gems could have red color. The bluish green variety of beryl is known as aquamarine or bhadra, and was mentioned in the Arthasastra as uptpalavarnah (like blue lotus). The Arthasastra also mentions several subsidiary types of gems named after their color, lustre or place of origin. Vimalaka shining pyrite, white-red jyotirasaka, (could be agate and carnelian), lohitaksa, black in the centre and red at the fringe (magnetite; and hematite on the fringe?), sasyaka blue copper sulphate, ahicchatraka from Ahicchatra, suktichurnaka powdered oyster, ksiravaka, milk coloured gem or lasuna and bukta pulaka (with chatoyancy or change in lustre) which could be cat's eye, a variety of chrysoberyl, and so on.
The authors further mention that at the end was mentioned kacamani, the amorphous gems or artificial gems imitated by coloring glass. The technique of maniraga or imparting colour to produce artificial gems was specifically mentioned.
We are told that the Arthasastra also mentions the uses of several non-gem mineral and materials such as pigments, mordants, abrasives, materials producing alkali, salts, bitumen, charcoal, husk, etc.
Pigments were in use such as anjan ,( antimony sulphide), manahsil ( red arsenic sulphide), haritala, (yellow arsenic sulphide) and hinguluka (mercuric sulphide), Kastsa (green iron sulphate) and sasyaka, blue copper sulphate. These minerals were used as coloring agents and later as mordants in dyeing clothes. Of great commercial importance were metallic ores from which useful metals were extracted. The Arthasastra did not provide the names of the constituent minerals beyond referring to them as dhatu of iron (Tiksnadhatu), copper, lead, etc.
Having reviewed the literary evidence the authors maintain that the Arthasastra is the earliest Indian text dealing with the mineralogical characteristics of metallic ores and other mineral-aggregate rocks. It recognizes ores in the earth, in rocks, or in liquid form, with excessive color, heaviness and often-strong smell and taste. A gold-bearing ore is also described. Similarly, the silver ore described in the Arthasastra seems to be a complex sulphide ore containing silver (colour of a conch-shell), camphor, vimalaka (pyrite?).
The Arthasastra describes the sources and the qualities of good grade gold and silver ores. Copper ores were stated to be 'heavy, greasy, tawny (chalcopyrite left exposed to air tarnishes), green (color of malachite), dark blue with yellowish tint (azurite), pale red or red (native copper). Lead ores were stated to be grayish black, like kakamecaka (this is the color of galena), yellow like pigeon bile, marked with white lines (quartz or calcite gangue minerals) and smelling like raw flesh (odour of sulphur). Iron ore was known to be greasy stone of pale red colour, or of the colour of the sinduvara flower (hematite). After describing the above metallic ores or dhatus of specific metals, the Arthasastra writes: In that case vaikrntaka metal must be iron itself which used to be produced by the South Indians starting from the magnetite ore. It is not certain whether vaikrntaka metal was nickel or magnetite based iron. Was it the beginning of the famous Wootz steel?
The Arthasastra mentions specific uses of various metals of which gold and silver receive maximum attention. The duties of suvarna-adhyaksah, the 'Superintendent of Gold, are defined. He was supposed to establish industrial outfits and employ sauvarnikas or goldsmiths, well versed in the knowledge of not only gold and silver, but also of the alloying elements such as copper and iron and of gems which had to be set in the gold and silver wares. Gold smelting was known as suvarnapaka. Various ornamental alloys could be prepared by mixing variable proportions of iron and copper with gold, silver and sveta tara or white silver which contained gold, silver and some coloring matter. Two parts of silver and one part of copper constituted triputaka. An alloy of equal parts of silver and iron was known as vellaka.
Gold plating (tvastrkarma) could be done on silver or copper. Lead, copper or silver objects were coated with a gold-leaf (acitakapatra) on one side or with a twin-leaf fixed with lac etc. Gold, silver or gems were embedded (pinka) in solid or hollow articles by pasting a thick pulp of gold, silver or gem particles and the cementing agents such as lac, vermilion, red lead on the object and then heating.
The Arthasastra also describes a system of coinage based on silver and copper. The masaka, half masaka, quarter masaka known as the kakani, and half kakani, copper coins (progressively lower weights) had the same composition, viz., one-quarter hardening alloy and the rest copper.
The Arthasastra specifies that the Director of Metals (lohadhyakasa) should establish factories for metals (other than gold and silver) viz., copper, lead, tin, vaikrntaka, arakuta or brass, vratta (steel), kamsa (bronze), tala (bell-metal) and loha (iron or simply metal), and the corresponding metal-wares. In the Vedic era, copper was known as lohayasa or red metal. Copper used to be alloyed with arsenic to produce tala or bell metal and with trapu or tin to produce bronze. Zinc in India must have started around 400 BC in Taxila. Zawar mines in Rajasthan also give similar evidence. Vaikrntaka has been referred to some times with vrata, which is identified by many scholars including Kangle, as steel. On the top of it, tiksna mentioned as iron, had its ore or dhatu, and the metal was used as an alloying component. Iron prepared from South Indian magnetite or vaikrantakadhatu was wrongly believed to be a different metal.

Amazing Science (Part 7)
Pages from the history of the Indian sub-continent:
Science and Mathematics in India

History of Mathematics in India
Indic Mathematics - India and the Scientific Revolution
By David Gray, PhD

Why, one might ask, did Europe take over thousand years to attain the
level of abstract mathematics achieved by Indians such as Aaryabhatta?
The answer appears to be that Europeans were trapped in the relatively
simplistic and concrete geometrical mathematics developed by the Greeks.
It was not until they had, via the Arabs, received, assimilated and accepted
the place-value system of enumeration developed in India that they were
able to free their minds from the concrete and develop more abstract
systems of thought. This development thus triggered the scientific and
information technology revolutions which swept Europe and, later, the world.
The role played by India in the development 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 it's greatest contributions to world civilization.

Pages from the history of the Indian sub-continent:
Science and Mathematics in India

In all early civilizations, the first expression of mathematical understanding appears in the form of counting systems. Numbers in very early societies were typically represented by groups of lines, though later different numbers came to be assigned specific numeral names and symbols (as in India) or were designated by alphabetic letters (such as in Rome). Although today, we take our decimal system for granted, not all ancient civilizations based their numbers on a ten-base system. In ancient Babylon, a sexagesimal (base 60) system was in use.
The Decimal System in Harappa
In India a decimal system was already in place during the Harappan period, as indicated by an analysis of Harappan weights and measures. Weights corresponding to ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500 have been identified, as have scales with decimal divisions. A particularly notable characteristic of Harappan weights and measures is their remarkable accuracy. A bronze rod marked in units of 0.367 inches points to the degree of precision demanded in those times. Such scales were particularly important in ensuring proper implementation of town planning rules that required roads of fixed widths to run at right angles to each other, for drains to be constructed of precise measurements, and for homes to be constructed according to specified guidelines. The existence of a gradated system of accurately marked weights points to the development of trade and commerce in Harappan society.

Mathematical Activity in the Vedic Period

In the Vedic period, records of mathematical activity are mostly to be found in Vedic texts associated with ritual activities. However, as in many other early agricultural civilizations, the study of arithmetic and geometry was also impelled by secular considerations. Thus, to some extent early mathematical developments in India mirrored the developments in Egypt, Babylon and China . The system of land grants and agricultural tax assessments required accurate measurement of cultivated areas. As land was redistributed or consolidated, problems of mensuration came up that required solutions. In order to ensure that all cultivators had equivalent amounts of irrigated and non-irrigated lands and tracts of equivalent fertility - individual farmers in a village often had their holdings broken up in several parcels to ensure fairness. Since plots could not all be of the same shape - local administrators were required to convert rectangular plots or triangular plots to squares of equivalent sizes and so on. Tax assessments were based on fixed proportions of annual or seasonal crop incomes, but could be adjusted upwards or downwards based on a variety of factors. This meant that an understanding of geometry and arithmetic was virtually essential for revenue administrators. Mathematics was thus brought into the service of both the secular and the ritual domains.
Arithmetic operations (Ganit) such as addition, subtraction, multiplication, fractions, squares, cubes and roots are enumerated in the Narad Vishnu Purana attributed to Ved Vyas (pre-1000 BC). Examples of geometric knowledge (rekha-ganit) are to be found in the Sulva-Sutras of Baudhayana (800 BC) and Apasthmaba (600 BC) which describe techniques for the construction of ritual altars in use during the Vedic era. It is likely that these texts tapped geometric knowledge that may have been acquired much earlier, possibly in the Harappan period. Baudhayana's Sutra displays an understanding of basic geometric shapes and techniques of converting one geometric shape (such as a rectangle) to another of equivalent (or multiple, or fractional) area (such as a square). While some of the formulations are approximations, others are accurate and reveal a certain degree of practical ingenuity as well as some theoretical understanding of basic geometric principles. Modern methods of multiplication and addition probably emerged from the techniques described in the Sulva-Sutras.
Pythagoras - the Greek mathematician and philosopher who lived in the 6th C B.C was familiar with the Upanishads and learnt his basic geometry from the Sulva Sutras. An early statement of what is commonly known as the Pythagoras theorem is to be found in Baudhayana's Sutra: The chord which is stretched across the diagonal of a square produces an area of double the size. A similar observation pertaining to oblongs is also noted. His Sutra also contains geometric solutions of a linear equation in a single unknown. Examples of quadratic equations also appear. Apasthamba's sutra (an expansion of Baudhayana's with several original contributions) provides a value for the square root of 2 that is accurate to the fifth decimal place. Apasthamba also looked at the problems of squaring a circle, dividing a segment into seven equal parts, and a solution to the general linear equation. Jain texts from the 6th C BC such as the Surya Pragyapti describe ellipses.
Modern-day commentators are divided on how some of the results were generated. Some believe that these results came about through hit and trial - as rules of thumb, or as generalizations of observed examples. Others believe that once the scientific method came to be formalized in the Nyaya-Sutras - proofs for such results must have been provided, but these have either been lost or destroyed, or else were transmitted orally through the Gurukul system, and only the final results were tabulated in the texts. In any case, the study of Ganit i.e mathematics was given considerable importance in the Vedic period. The Vedang Jyotish (1000 BC) includes the statement: "Just as the feathers of a peacock and the jewel-stone of a snake are placed at the highest point of the body (at the forehead), similarly, the position of Ganit is the highest amongst all branches of the Vedas and the Shastras."
(Many centuries later, Jain mathematician from Mysore, Mahaviracharya further emphasized the importance of mathematics: "Whatever object exists in this moving and non-moving world, cannot be understood without the base of Ganit (i.e. mathematics)".)


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 Normal 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.
Philosophy and Mathematics
Philosophical doctrines also had a profound influence on the development of mathematical concepts and formulations. Like the Upanishadic world view, space and time were considered limitless in Jain cosmology. This led to a deep interest in very large numbers and definitions of infinite numbers. Infinite numbers were created through recursive formulae, as in the Anuyoga Dwara Sutra. Jain mathematicians recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite. Permutations and combinations are listed in the Bhagvati Sutras (3rd C BC) and Sathananga Sutra (2nd C BC).
Jain set theory probably arose in parallel with the Syadvada system of Jain epistemology in which reality was described in terms of pairs of truth conditions and state changes. The Anuyoga Dwara Sutra demonstrates an understanding of the law of indeces and uses it to develop the notion of logarithms. Terms like Ardh Aached , Trik Aached, and Chatur Aached are used to denote log base 2, log base 3 and log base 4 respectively. In Satkhandagama various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers. The operations are repeated to produce new sets. In other works the relation of the number of combinations to the coefficients occurring in the binomial expansion is noted.
Since Jain epistemology allowed for a degree of indeterminacy in describing reality, it probably helped in grappling with indeterminate equations and finding numerical approximations to irrational numbers.
Buddhist literature also demonstrates an awareness of indeterminate and infinite numbers. Buddhist mathematics was classified either as Garna (Simple Mathematics) or Sankhyan (Higher Mathematics). Numbers were deemed to be of three types: Sankheya (countable), Asankheya (uncountable) and Anant (infinite).
Philosophical formulations concerning Shunya - i.e. emptiness or the void may have facilitated in the introduction of the concept of zero. While the zero (bindu) as an empty place holder in the place-value numeral system appears much earlier, algebraic definitions of the zero and it's relationship to mathematical functions appear in the mathematical treatises of Brahmagupta in the 7th C AD. Although scholars are divided about how early the symbol for zero came to be used in numeric notation in India, (Ifrah arguing that the use of zero is already implied in Aryabhatta) tangible evidence for the use of the zero begins to proliferate towards the end of the Gupta period. Between the 7th C and the 11th C, Indian numerals developed into their modern form, and along with the symbols denoting various mathematical functions (such as plus, minus, square root etc) eventually became the foundation stones of modern mathematical notation.
The Indian Numeral System
Although the Chinese were also using a decimal based counting system, the Chinese lacked a formal notational system that had the abstraction and elegance of the Indian notational system, and it was the Indian notational system that reached the Western world through the Arabs and has now been accepted as universal. Several factors contributed to this development whose significance is perhaps best stated by French mathematician, Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. It's simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions."
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.
Influence of Trade and Commerce, Importance of Astronomy
The growth of trade and commerce, particularly lending and borrowing demanded an understanding of both simple and compound interest which probably stimulated the interest in arithmetic and geometric series. Brahmagupta's description of negative numbers as debts and positive numbers as fortunes points to a link between trade and mathematical study. Knowledge of astronomy - particularly knowledge of the tides and the stars was of great import to trading communities who crossed oceans or deserts at night. This is borne out by numerous references in the Jataka tales and several other folk-tales. The young person who wished to embark on a commercial venture was inevitably required to first gain some grounding in astronomy. This led to a proliferation of teachers of astronomy, who in turn received training at universities such as at Kusumpura (Bihar) or Ujjain (Central India) or at smaller local colleges or Gurukuls. This also led to the exchange of texts on astronomy and mathematics amongst scholars and the transmission of knowledge from one part of India to another. Virtually every Indian state produced great mathematicians who wrote commentaries on the works of other mathematicians (who may have lived and worked in a different part of India many centuries earlier). Sanskrit served as the common medium of scientific communication.
The science of astronomy was also spurred by the need to have accurate calendars and a better understanding of climate and rainfall patterns for timely sowing and choice of crops. At the same time, religion and astrology also played a role in creating an interest in astronomy and a negative fallout of this irrational influence was the rejection of scientific theories that were far ahead of their time. One of the greatest scientists of the Gupta period - Aryabhatta (born in 476 AD, Kusumpura, Bihar) provided a systematic treatment of the position of the planets in space. He correctly posited the axial rotation of the earth, and inferred correctly that the orbits of the planets were ellipses. He also correctly deduced that the moon and the planets shined by reflected sunlight and provided a valid explanation for the solar and lunar eclipses rejecting the superstitions and mythical belief systems surrounding the phenomenon. Although Bhaskar I (born Saurashtra, 6th C, and follower of the Asmaka school of science, Nizamabad, Andhra ) recognized his genius and the tremendous value of his scientific contributions, some later astronomers continued to believe in a static earth and rejected his rational explanations of the eclipses. But in spite of such setbacks, Aryabhatta had a profound influence on the astronomers and mathematicians who followed him, particularly on those from the Asmaka school.
Mathematics played a vital role in Aryabhatta's revolutionary understanding of the solar system. His calculations on pi, the circumferance of the earth (62832 miles) and the length of the solar year (within about 13 minutes of the modern calculation) were remarkably close approximations. In making such calculations, Aryabhatta had to solve several mathematical problems that had not been addressed before, including problems in algebra (beej-ganit) and trigonometry (trikonmiti).
Bhaskar I continued where Aryabhatta left off, and discussed in further detail topics such as the longitudes of the planets; conjunctions of the planets with each other and with bright stars; risings and settings of the planets; and the lunar crescent. Again, these studies required still more advanced mathematics and Bhaskar I expanded on the trigonometric equations provided by Aryabhatta, and like Aryabhatta correctly assessed pi to be an irrational number. Amongst his most important contributions was his formula for calculating the sine function which was 99% accurate. He also did pioneering work on indeterminate equations and considered for the first time quadrilaterals with all the four sides unequal and none of the opposite sides parallel.
Another important astronomer/mathematician was Varahamira (6th C, Ujjain) who compiled previously written texts on astronomy and made important additions to Aryabhatta's trigonometric formulas. His works on permutations and combinations complemented what had been previously achieved by Jain mathematicians and provided a method of calculation of nCr that closely resembles the much more recent Pascal's Triangle. In the 7th century, Brahmagupta did important work in enumerating the basic principles of algebra. In addition to listing the algebraic properties of zero, he also listed the algebraic properties of negative numbers. His work on solutions to quadratic indeterminate equations anticipated the work of Euler and Lagrange.
Emergence of Calculus
In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was obliged to introduce the concept of infinitesimals - i.e. tatkalika gati to designate the infinitesimal, or near instantaneous motion of the moon, and express it in the form of a basic differential equation. Aryabhatta's equations were elaborated on by Manjula (10th C) and Bhaskaracharya (12th C) who derived the differential of the sine function. Later mathematicians used their intuitive understanding of integration in deriving the areas of curved surfaces and the volumes enclosed by them.
Applied Mathematics, Solutions to Practical Problems
Developments also took place in applied mathematics such as in creation of trigonometric tables and measurement units. Yativrsabha's work Tiloyapannatti (6th C) gives various units for measuring distances and time and also describes the system of infinite time measures.
In the 9th C, Mahaviracharya ( Mysore) wrote Ganit Saar Sangraha where he described the currently used method of calculating the Least Common Multiple (LCM) of given numbers. He also derived formulae to calculate the area of an ellipse and a quadrilateral inscribed within a circle (something that had also been looked at by Brahmagupta) The solution of indeterminate equations also drew considerable interest in the 9th century, and several mathematicians contributed approximations and solutions to different types of indeterminate equations.
In the late 9th C, Sridhara (probably Bengal) provided mathematical formulae for a variety of practical problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns. Some of these examples involved fairly complicated solutions and his Patiganita is considered an advanced mathematical work. Sections of the book were also devoted to arithmetic and geometric progressions, including progressions with fractional numbers or terms, and formulas for the sum of certain finite series are provided. Mathematical investigation continued into the 10th C. Vijayanandi (of Benares, whose Karanatilaka was translated by Al-Beruni into Arabic) and Sripati of Maharashtra are amongst the prominent mathematicians of the century.
The leading light of 12th C Indian mathematics was Bhaskaracharya who came from a long-line of mathematicians and was head of the astronomical observatory at Ujjain. He left several important mathematical texts including the Lilavati and Bijaganita and the Siddhanta Shiromani, an astronomical text. He was the first to recognize that certain types of quadratic equations could have two solutions. His Chakrawaat method of solving indeterminate solutions preceded European solutions by several centuries, and in his Siddhanta Shiromani he postulated that the earth had a gravitational force, and broached the fields of infinitesimal calculation and integration. In the second part of this treatise, there are several chapters relating to the study of the sphere and it's properties and applications to geography, planetary mean motion, eccentric epicyclical model of the planets, first visibilities of the planets, the seasons, the lunar crescent etc. He also discussed astronomical instruments and spherical trigonometry. Of particular interest are his trigonometric equations: sin(a + b) = sin a cos b + cos a sin b; sin(a - b) = sin a cos b - cos a sin b;
The Spread of Indian Mathematics
The study of mathematics appears to slow down after the onslaught of the Islamic invasions and the conversion of colleges and universities to madrasahs. But this was also the time when Indian mathematical texts were increasingly being translated into Arabic and Persian. Although Arab scholars relied on a variety of sources including Babylonian, Syrian, Greek and some Chinese texts, Indian mathematical texts played a particularly important role. Scholars such as Ibn Tariq and Al-Fazari (8th C, Baghdad), Al-Kindi (9th C, Basra), Al-Khwarizmi (9th C. Khiva), Al-Qayarawani (9th C, Maghreb, author of Kitab fi al-hisab al-hindi), Al-Uqlidisi (10th C, Damascus, author of The book of Chapters in Indian Arithmetic), Ibn-Sina (Avicenna), Ibn al-Samh (Granada, 11th C, Spain), Al-Nasawi (Khurasan, 11th C, Persia), Al-Beruni (11th C, born Khiva, died Afghanistan), Al-Razi (Teheran), and Ibn-Al-Saffar (11th C, Cordoba) were amongst the many who based their own scientific texts on translations of Indian treatises. Records of the Indian origin of many proofs, concepts and formulations were obscured in the later centuries, but the enormous contributions of Indian mathematics was generously acknowledged by several important Arabic and Persian scholars, especially in Spain. Abbasid scholar Al-Gaheth wrote: " India is the source of knowledge, thought and insight”. Al-Maoudi (956 AD) who travelled in Western India also wrote about the greatness of Indian science. Said Al-Andalusi, an 11th C Spanish scholar and court historian was amongst the most enthusiastic in his praise of Indian civilization, and specially remarked on Indian achievements in the sciences and in mathematics. Of course, eventually, Indian algebra and trigonometry reached Europe through a cycle of translations, travelling from the Arab world to Spain and Sicily, and eventually penetrating all of Europe. At the same time, Arabic and Persian translations of Greek and Egyptian scientific texts became more readily available in India
The Kerala School
Although it appears that original work in mathematics ceased in much of Northern India after the Islamic conquests, Benaras survived as a center for mathematical study, and an important school of mathematics blossomed in Kerala. Madhava (14th C, Kochi) made important mathematical discoveries that would not be identified by European mathematicians till at least two centuries later. His series expansion of the cos and sine functions anticipated Newton by almost three centuries. Historians of mathematics, Rajagopal, Rangachari and Joseph considered his contributions instrumental in taking mathematics to the next stage, that of modern classical analysis. Nilkantha (15th C, Tirur, Kerala) extended and elaborated upon the results of Madhava while Jyesthadeva (16th C, Kerala) provided detailed proofs of the theorems and derivations of the rules contained in the works of Madhava and Nilkantha. It is also notable that Jyesthadeva's Yuktibhasa which contained commentaries on Nilkantha's Tantrasamgraha included elaborations on planetary theory later adopted by Tycho Brahe, and mathematics that anticipated work by later Europeans. Chitrabhanu (16th C, Kerala) gave integer solutions to twenty-one types of systems of two algebraic equations, using both algebraic and geometric methods in developing his results. Important discoveries by the Kerala mathematicians included the Newton-Gauss interpolation formula, the formula for the sum of an infinite series, and a series notation for pi. Charles Whish (1835, published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland) was one of the first Westerners to recognize that the Kerala school had anticipated by almost 300 years many European developments in the field.
Yet, few modern compendiums on the history of mathematics have paid adequate attention to the often pioneering and revolutionary contributions of Indian mathematicians. But as this essay amply demonstrates, a significant body of mathematical works were produced in the Indian subcontinent. The science of mathematics played a pivotal role not only in the industrial revolution but in the scientific developments that have occurred since. No other branch of science is complete without mathematics. Not only did India provide the financial capital for the industrial revolution (see the essay on colonization) India also provided vital elements of the scientific foundation without which humanity could not have entered this modern age of science and high technology.
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.

Mathematics and Architecture: Interest in arithmetic and geometric series may have also been stimulated by (and influenced) Indian architectural designs - (as in temple shikaras, gopurams and corbelled temple ceilings). Of course, the relationship between geometry and architectural decoration was developed to it's greatest heights by Central Asian, Persian, Turkish, Arab and Indian architects in a variety of monuments commissioned by the Islamic rulers.
Transmission of the Indian Numeral System: Evidence for the transmission of the Indian Numeral System to the West is provided by Joseph (Crest of the Peacock):-
· Quotes Severus Sebokht (662) in a Syriac text describing the "subtle discoveries" of Indian astronomers as being "more ingenious than those of the Greeks and the Babylonians" and "their valuable methods of computation which surpass description" and then goes on to mention the use of nine numerals.

· Quotes from Liber abaci (Book of the Abacus) by Fibonacci (1170-1250): The nine Indian numerals are ...with these nine and with the sign 0 which in Arabic is sifr, any desired number can be written. (Fibonaci learnt about Indian numerals from his Arab teachers in North Africa)

Influence of the Kerala School: Joseph (Crest of the Peacock) suggests that Indian mathematical manuscripts may have been brought to Europe by Jesuit priests such as Matteo Ricci who spent two years in Kochi (Cochin) after being ordained in Goa in 1580. Kochi is only 70km from Thrissur (Trichur) which was then the largest repository of astronomical documents. Whish and Hyne - two European mathematicians obtained their copies of works by the Kerala mathematicians from Thrissur, and it is not inconceivable that Jesuit monks may have also taken copies to Pisa (where Galileo, Cavalieri and Wallis spent time), or Padau (where James Gregory studied) or Paris (where Mersenne who was in touch with Fermat and Pascal, acted as an agent for the transmission of mathematical ideas).
1.Studies in the History of Science in India(Anthology edited by Debiprasad Chattopadhyaya)
2.A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin: Studies in the history of mathematics, "Nauka" (Moscow, 1974), 220-222; 302.

3. B Datta: The science of the Sulba (Calcutta, 1932).
4.G G Joseph:The crest of the peacock (Princeton University Press, 2000).
5. R P Kulkarni: The value of pi known to Sulbasutrakaras, Indian Journal Hist. Sci. 13 (1) (1978), 32-41.
6. G Kumari:Some significant results of algebra of pre-Aryabhata era, Math. Ed. (Siwan) 14 (1) (1980), B5-B13.
7. G Ifrah: A universal history of numbers: From prehistory to the invention of the computer (London, 1998).
8. P Z Ingerman: 'Panini-Backus form', Communications of the ACM 10 (3)(1967), 137.
9.P Jha, Contributions of the Jainas to astronomy and mathematics, Math. Ed. (Siwan) 18 (3) (1984), 98-107.

9b. R C Gupta: The first unenumerable number in Jaina mathematics, Ganita Bharati 14 (1-4) (1992), 11-24.
10. L C Jain: System theory in Jaina school of mathematics, Indian J. Hist. Sci. 14 (1) (1979), 31-65.
11. L C Jain and Km Meena Jain: System theory in Jaina school of mathematics. II, Indian J. Hist. Sci. 24 (3) (1989), 163-180
12. K Shankar Shukla: Bhaskara I, Bhaskara I and his works II. Maha-Bhaskariya (Sanskrit) (Lucknow, 1960).
13. K Shankar Shukla: Bhaskara I, Bhaskara I and his works III. Laghu-Bhaskariya (Sanskrit) (Lucknow, 1963).
14. K S Shukla: Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya, Ganita 22 (1) (1971), 115-130.
15. R C Gupta: Varahamihira's calculation of nCr and the discovery of Pascal's triangle, Ganita Bharati 14 (1-4) (1992), 45-49.
16. B Datta: On Mahavira's solution of rational triangles and quadrilaterals, Bull. Calcutta Math. Soc. 20 (1932), 267-294.
17. B S Jain: On the Ganita-Sara-Samgraha of Mahavira (c. 850 A.D.), Indian J. Hist. Sci. 12 (1) (1977), 17-32.
18. K Shankar Shukla: The Patiganita of Sridharacarya (Lucknow, 1959).
19. H. Suter: Mathematiker
20. Suter: Die Mathematiker und Astronomen der Araber
21. Die philosophischen Abhandlungen des al-Kindi, Munster, 1897
22. K V Sarma: A History of the Kerala School of Hindu Astronomy (Hoshiarpur, 1972).
23. R C Gupta: The Madhava-Gregory series, Math. Education 7 (1973), B67-B70
24. S Parameswaran: Madhavan, the father of analysis, Ganita-Bharati 18 (1-4) (1996), 67-70.
25. K V Sarma, and S Hariharan: Yuktibhasa of Jyesthadeva : a book of rationales in Indian mathematics and astronomy - an analytical appraisal, Indian J. Hist. Sci. 26 (2) (1991), 185-207
26. C T Rajagopal and M S Rangachari: On an untapped source of medieval Keralese mathematics, Arch. History Exact Sci. 18 (1978), 89-102.
27. C T Rajagopal and M S Rangachari: On medieval Keralese mathematics, Arch. History Exact Sci. 35 (1986), 91-99.
28. A.K. Bag: Mathematics in Ancient and Medieval India (1979, Varanasi)
29. Bose, Sen, Subarayappa: Concise History of Science in India, (Indian National Science Academy)
30. T.A. Saraswati: Geometry in Ancient and Medieval India (1979, Delhi)
31.N. Singh: Foundations of Logic in Ancient India, Linguistics and Mathematics ( Science and technology in Indian Culture, ed. A Rahman, 1984, New Delhi, National Instt. of Science, Technology and Development Studies, NISTAD)
32. P. Singh: "The so-called Fibonacci numbers in ancient and medieval India, (Historia Mathematica, 12, 229-44, 1985)
33. Chin Keh-Mu: India and China: Scientific Exchange (History of Science in India Vol 2.)
Another view on Indian Mathematics:
Indic Mathematics and the Scientific Revolution
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."
      -Indic Mathematics

India and the Scientific Revolution
By David Gray, PhD
1. Math and Ethnocentrism
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. This awakening was in part made possible by the rediscovery of mathematics and other sciences and technologies through the medium of the Arabs, who transmitted to Europe both their own lost heritage as well as the advanced mathematical traditions formulated in India.
George Ghevarughese Joseph, in an important article entitled "Foundations of Eurocentrism in Mathematics," argued that "the standard treatment of the history of non-European mathematics is a product of historiographical bias (conscious or otherwise) in the selection and interpretation of facts, which, as a consequence, results in ignoring, devaluing or distorting contributions arising outside European mathematical traditions." (1987:14)
Due to the legacy of colonialism, the exploitation of which was ideologically justified through a doctrine of racial superiority, the contributions of non-European civilizations were often ignored, or, as Joseph argued, even distorted, in that they were often misattributed as European, i.e. Greek, contributions, and when their contributions were so great as to resist such treatment, they were typically devalued, considered inferior or irrelevant to Western mathematical traditions.
This tendency has not only led to the devaluation of non-Western mathematical traditions, but has distorted the history of Western mathematics as well. In so far as the contributions from non-Western civilizations are ignored, there is the problem of accounting for the development of mathematics purely within the Western cultural framework. This has led, as Sabetai Unguru has argued, toward a tendency to read more advanced mathematical concepts into the relatively simplistic geometrical formulations of Greek mathematicians such as Euclid, despite the fact that the Greeks lacked not only mathematic notation, but even the place-value system of enumeration, without which advanced mathematical calculation is impossible. Such ethnocentric revisionist history resulted in the attribution of more advanced algebraic concepts, which were actually introduced to Europe over a millennium later by the Arabs, to the Greeks. And while the contributions of the Greeks to mathematics was quite significant, the tendency of some math historians to jump from the Greeks to renaissance Europe results not only in an ethnocentric history, but an inadequate history as well, one which fails to take into account the full history of the development of modern mathematics, which is by no means a purely European development.
2. Vedic Altars and the "Pythagorean theorem"
A perfect example of this sort of misattribution involves the so-called Pythagorean theorem, the well-known theorem which was attributed to Pythagoras who lived around 500 BCE, but which was first proven in Greek sources in Euclid's Geometry, written centuries later. Despite the scarcity of evidence backing this attribution, it is not often questioned, perhaps due to the mantra-like frequency with which it is repeated. However, Seidenberg, in his 1978 article, shows that the thesis that Greece was the origin of geometric algebra was incorrect, "for geometric algebra existed in India before the classical period in Greece." (1978:323) It is now generally understood that the so-called "Pythagorean theorem" was understood in ancient India, and was in fact proved in Baudhayana's Shulva Sutra, a text dated to circa 600 BCE. (1978:323).
Knowledge of mathematics, and geometry in particular, was necessary for the precise construction of the complex Vedic altars, and mathematics was thus one of the topics covered in the brahmanas. This knowledge was further elaborated in the kalpa sutras, which gave more detailed instructions concerning Vedic ritual. Several of these treat the topic of altar construction. The oldest and most complete of these is the previously mentioned Shulva Sutra of Baudhaayana. As this text was composed about a century before Pythagoras, the theory that the Greeks were the source of Geometric algebra is untenable, while the hypothesis that India was have been a source for Greek geometry, transmitted via the Persians who traded both with the Greeks and the Indians, looks increasingly plausible. On the other hand, it is quite possible that both the Greeks and the Indians developed geometry. Seidenberg has argued, in fact, that both seem to have developed geometry out of the practical problems involving their construction of elaborate sacrificial altars. (See Seidenberg 1962 and 1983)
3. Zero and the Place Value System
Far more important to the development of modern mathematics than either Greek or Indian geometry was the development of the place value system of enumeration, the base ten system of calculation which uses nine numerals and zero to represent numbers ranging from the most minuscule decimal to the most inconceivably large power of ten. This system of enumeration was not developed by the Greeks, whose largest unit of enumeration was the myriad (10,000) or in China, where 10,000 was also the largest unit of enumeration until recent times. Nor was it developed by the Arabs, despite the fact that this numeral system is commonly called the Arabic numerals in Europe, where this system was first introduced by the Arabs in the thirteenth century.
Rather, this system was invented in India, where it evidently was of quite ancient origin. The Yajurveda Samhitaa, one of the Vedic texts predating Euclid and the Greek mathematicians by at least a millennium, lists names for each of the units of ten up to 10 to the twelfth power (paraardha). (Subbarayappa 1970:49) Later Buddhist and Jain authors extended this list as high as the fifty-third power, far exceeding their Greek contemporaries, who lacking a system of enumeration were unable to develop abstract mathematical concepts.
The place value system of enumeration is in fact built into the Sanskrit language, where each power of ten is given a distinct name. Not only are the units ten, hundred and thousand (daza, zata, sahasra) named as in English, but also ten thousand, hundred thousand, ten million, hundred million (ayuta, lakSa, koti, vyarbuda), and so forth up to the fifty-third power, providing distinct names where English makes use of auxillary bases such as thousand, million, etc. Ifrah has commented that
By giving each power of ten an individual name, the Sanskrit system gave no special importance to any number. Thus the Sanskrit system is obviously superior to that of the Arabs (for whom the thousand was the limit), or the Greeks and Chinese (whose limit was ten thousand) and even to our own system (where the names thousand, million etc. continue to act as auxillary bases). Instead of naming the numbers in groups of three, four or eight orders of units, the Indians, from a very early date, expressed them taking the powers of ten and the names of the first nine units individually. In other words, to express a given number, one only had to place the name indicating the order of units between the name of the order of units immediately immediately below it and the one immediately above it. That is exactly what is required in order to gain a precise idea of the place-value system, the rule being presented in a natural way and thus appearing self-explanatory. To put it plainly, the Sanskrit numeral system contained the very key to the discovery of the place-value system. (2000:429)
As Ifrah has shown at length, there is little doubt that our place-value numeral system developed in India (2000:399-409), and this system is embedded in the Sanskrit language, several aspects of which make it a very logical language, well suited to scientific and mathematical reasoning. Nor did this system exhaust Indian ingenuity; as van Nooten has shown, Pingala, who lived circa the first century BCE, developed a system of binary enumeration convertible to decimal numerals, described in his Chandahzaastra. His system is quite similar to that of Leibniz, who lived roughly fourteen hundred years later. (See Van Nooten)
India is also the locus of another closely related an equally important mathematical discovery, the numeral zero. The oldest known text to use zero is a Jain text entitled the Lokavibhaaga, which has been definitely dated to Monday 25 August 458 CE. (Ifrah 2000:417-1 9) This concept, combined by the place-value system of enumeration, became the basis for a classical era renaissance in Indian mathematics.
The Indian numeral system and its place value, decimal system of enumeration came to the attention of the Arabs in the seventh or eighth century, and served as the basis for the well known advancement in Arab mathematics, represented by figures such as al-Khwarizmi. It reached Europe in the twelfth century when Adelard of Bath translated al-Khwarizmi's works into Latin. (Subbarayappa 1970:49) But the Europeans were at first resistant to this system, being attached to the far less logical roman numeral system, but their eventual adoption of this system led to the scientific revolution that began to sweep Europe beginning in the thirteenth century.
4. Luminaries of Classical Indian Mathematics
The world did not have to wait for the Europeans to awake from their long intellectual slumber to see the development of advanced mathematical techniques. India achieved its own scientific renaissance of sorts during its classical era, beginning roughly one thousand years before the European Renaissance. Probably the most celebrated Indian mathematicians belonging to this period was Aaryabhat.a, who was born in 476 CE.
In 499, when he was only 23 years old, Aaryabhat.a wrote his Aaryabhatiya, a text covering both astronomy and mathematics. With regard to the former, the text is notable for its for its awareness of the relativity of motion. (See Kak p. 16) This awareness led to the astonishing suggestion that it is the Earth that rotates the Sun. He argued for the diurnal rotation of the earth, as an alternate theory to the rotation of the fixed stars and sun around the earth (Pingree 1981:18). He made this suggestion approximately one thousand years before Copernicus, evidently independently, reached the same conclusion.

With regard to mathematics, one of Aaryabhat.a's greatest contributions was the calculation of sine tables, which no doubt was of great use for his astronomical calculations. In developing a way to calculate the sine of curves, rather than the cruder method of calculating chords devised by the Greeks, he thus went beyond geometry and contributed to the development of trigonometry, a development which did not occur in Europe until roughly one thousand years later, when the Europeans translated Indian influenced Arab mathematical texts.
Aaryabhat.a's mathematics was far ranging, as the topics he covered include geometry, algebra, trigonometry. He also developed methods of solving quadratic and indeterminate equations using fractions. He calculated pi to four decimal places, i.e., 3.1416. (Pingree 1981:57) In addition, Aaryabhat.a "invented a unique method of recording numbers which required perfect understanding of zero and the place-value system." (Ifrah 2000:419)
Given the astounding range of advanced mathematical concepts and techniques covered in this fifth century text, it should be of no surprise that it became extremely well known in India, judging by the large numbers of commentaries written upon it. It was studied by the Arabs in the eighth century following their conquest of Sind, and translated into Arabic, whence it influenced the development of both Arabic and European mathematical traditions.
Article by: J J O'Connor and E F Robertson:
The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines. Let us examine some of these in a little more detail.
First we look at the system for representing numbers which Aryabhata invented and used in the Aryabhatiya. It consists of giving numerical values to the 33 consonants of the Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000, .... In fact the system allows numbers up to 1018 to be represented with an alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar with numeral symbols and the place-value system. He writes in [3]:-
... it is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.
Next we look briefly at some algebra contained in the Aryabhatiya. This work is the first we are aware of which examines integer solutions to equations of the form by = ax + c and by = ax - c, where a, b, c are integers. The problem arose from studying the problem in astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to solve problems of this type. The word kuttaka means "to pulverise" and the method consisted of breaking the problem down into new problems where the coefficients became smaller and smaller with each step. The method here is essentially the use of the Euclidean algorithm to find the highest common factor of a and b but is also related to continued fractions.
Aryabhata gave an accurate approximation for p. He wrote in the Aryabhatiya the following:-
Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.
This gives p = 62832/20000 = 3.1416 which is a surprisingly accurate value. In fact p = 3.14159265 correct to 8 places. If obtaining a value this accurate is surprising, it is perhaps even more surprising that Aryabhata does not use his accurate value for p but prefers to use 10 = 3.1622 in practice. Aryabhata does not explain how he found this accurate value but, for example, Ahmad [5] considers this value as an approximation to half the perimeter of a regular polygon of 256 sides inscribed in the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling of the number of sides. Another interesting paper discussing this accurate value of p by Aryabhata is [22] where Jha writes:-
Aryabhata I's value of p is a very close approximation to the modern value and the most accurate among those of the ancients. There are reasons to believe that Aryabhata devised a particular method for finding this value. It is shown with sufficient grounds that Aryabhata himself used it, and several later Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of p is of Greek origin is critically examined and is found to be without foundation. Aryabhata discovered this value independently and also realised that p is an irrational number. He had the Indian background, no doubt, but excelled all his predecessors in evaluating p. Thus the credit of discovering this exact value of p may be ascribed to the celebrated mathematician, Aryabhata I.
We now look at the trigonometry contained in Aryabhata's treatise. He gave a table of sines calculating the approximate values at intervals of 90 /24 = 3 45'. In order to do this he used a formula for sin(n+1)x - sin nx in terms of sin nx and sin (n-1)x. He also introduced the versine (versin = 1 - cosine) into trigonometry.
Other rules given by Aryabhata include that for summing the first n integers, the squares of these integers and also their cubes. Aryabhata gives formulas for the areas of a triangle and of a circle which are correct, but the formulas for the volumes of a sphere and of a pyramid are claimed to be wrong by most historians. For example Ganitanand in [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V = Ah/2 for the volume of a pyramid with height h and triangular base of area A. He also appears to give an incorrect expression for the volume of a sphere. However, as is often the case, nothing is as straightforward as it appears and Elfering (see for example [13]) argues that this is not an error but rather the result of an incorrect translation.
This relates to verses 6, 7, and 10 of the second section of the Aryabhatiya and in [13] Elfering produces a translation which yields the correct answer for both the volume of a pyramid and for a sphere. However, in his translation Elfering translates two technical terms in a different way to the meaning which they usually have. Without some supporting evidence that these technical terms have been used with these different meanings in other places it would still appear that Aryabhata did indeed give the incorrect formulas for these volumes.
Now we have looked at the mathematics contained in the Aryabhatiya but this is an astronomy text so we should say a little regarding the astronomy which it contains. Aryabhata gives a systematic treatment of the position of the planets in space. He gave 62832 miles as the circumference of the earth, which is an excellent approximation. He believed that the apparent rotation of the heavens was due to the axial rotation of the Earth. This is a quite remarkable view of the nature of the solar system which later commentators could not bring themselves to follow and most changed the text to save Aryabhata from what they thought were stupid errors!
Aryabhata gives the radius of the planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sun. He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses of the Sun and the Moon. The Indian belief up to that time was that eclipses were caused by a demon called Rahu. His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours. Bhaskara
who wrote a commentary on the Aryabhatiya about 100 years later wrote of Aryabhata:-
Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world.
Article by: J J O'Connor and E F Robertson
Born in 598 CE in Rajastan in Western India, Brahmagupta founded an influential school of mathematics which rivaled Aaryabhat.a's. His best known work is the Brahmasphuta Siddhanta, written in 628 CE, in which he developed a solution for a certain type of second order indeterminate equation. This text was translated into Arabic in the eighth century, and became very influential in Arab mathematics. (See Kak p. 16)
Mahaaviira was a Jain mathematician who lived in the ninth century, who wrote on a wide range of mathematical topics. These include the mathematics of zero, squares, cubes, square-roots, cube-roots, and the series extending beyond these. He also wrote on plane and solid geometry, as well as problems relating to the casting of shadows. (Pingree 1981:60)
Bhaaskara was one of the many outstanding mathematicians hailing from South India. Born in 1114 CE in Karnataka, he composed a four-part text entitled the Siddhanta Ziromani. Included in this compilation is the Biijagan.ita, which became the standard algebra textbook in Sanskrit. It contains descriptions of advanced mathematical techniques involving both positive and negative integers as well as zero, irrational numbers. It treats at length the "pulverizer" (kut.t.akaara) method of solving indeterminate equations with continued fractions, as well as the so-called "Pell's equation (vargaprakr.ti) dealing with indeterminate equations of the second degree. He also wrote on the solution to numerous kinds of linear and quadratic equations, including those involving multiple unknowns, and equations involving the product of different unknowns. (Pingree 1981, p. 64)
In short, he wrote a highly sophisticated mathematical text that proceeded by several centuries the development of such techniques in Europe, although it would be better to term this a rediscovery, since much of the Renaissance advances of mathematics in Europe was based upon the discovery of Arab mathematical texts, which were in turn highly influenced by these Indian traditions.
The Kerala region of South India was home to a very important school of mathematics. The best known member of this school Maadhava (c. 1444-1545), who lived in Sangamagraama in Kerala. Primarily an astronomer, he made history in mathematics with his writings on trigonometry. He calculated the sine, cosine and arctangent of the circle, developing the world's first consistent system of trigonometry. (See Hayashi 1997:784-786) He also correctly calculated the value of p to eleven decimal places. (Pingree 1981:490)
This is by no means a complete list of influential Indian mathematicians or Indian contributions to mathematics, but rather a survey of the highlights of what is, judged by any fair, unbiased standard, an illustrious tradition, important both for its own internal elegance as well as its influence on the history of European mathematical traditions. The classical Indian mathematical renaissance was an important precursor to the European renaissance, and to ignore this fact is to fail to grasp the history of latter, a history which was truly multicultural, deriving its inspiration from a variety of cultural roots.
There are in fact, as Frits Staal has suggested in his important (1995) article, "The Sanskrit of Science", profound similarities between the social contexts of classical India and renaissance Europe. In both cases, important revolutions in scientific thought occurred in complex, hierarchical societies in which certain elite groups were granted freedom from manual labor, and thus the opportunity to dedicate themselves to intellectual pursuits. In the case of classical India, these groups included certain brahmins as well as the Buddhist and Jain monks, while in renaissance Europe they included both the monks as well as their secular derivatives, the university scholars.
Why, one might ask, did Europe's take over thousand years to attain the level of abstract mathematics achieved by Indians such as Aaryabhatta? The answer appears to be that Europeans were trapped in the relatively simplistic and concrete geometrical mathematics developed by the Greeks. It was not until they had, via the Arabs, received, assimilated and accepted the place-value system of enumeration developed in India that they were able to free their minds from the concrete and develop more abstract systems of thought. This development thus triggered the scientific and information technology revolutions which swept Europe and, later, the world. The role played by India in the development 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 it's greatest contributions to world civilization.
Works Cited
Hayashi, Takao. 1997. "Number Theory in India". In Helaine Selin, ed. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Boston: Kluwer Academic Publishers, pp. 784-786.
Ifrah, Georges. 2000. The Universal History of Numbers: From Prehistory to the Invention of the Computer. David Bellos, E. F. Harding, Sophie Wood and Ian Monk, trans. New York: John Wiley & Sons, Inc.
Joseph, George Ghevarughese. 1987. "Foundations of Eurocentrism in Mathematics". In Race & Class 28.3, pp. 13-28.
Kak, Subhash. "An Overview of Ancient Indian Science". In T. R. N. Rao and Subhash Kak, eds. Computing Science in Ancient India, pp. 6-21.
van Nooten, B. "Binary Numbers in Indian Antiquity". In T. R. N. Rao and Subhash Kak, eds. Computing Science in Ancient India, pp. 21-39.
Pingree, David. Jyotih.zaastra: Astral and Mathematical Literature, Wiesbaden: Otto Harrassowitz, 1981, p. 4.
Seidenberg, A. 1962. "The Ritual Origin of Geometry". In Archive for History of Exact Sciences 1, pp. 488-527.
______. 1978. "The Origin of Mathematics". In Archive for History of Exact Sciences 18.4, pp. 301-42.
______. 1983. "The Geometry of Vedic Rituals". In Frits Staal, ed. Agni: The Vedic Ritual of the Fire Altar. Delhi: Motilal Banarsidass, 1986, vol. 2, pp. 95-126.
Unguru, Sabetai. 1975. "On the Need to Rewrite the History of Greek Mathematics". In Archive for History of Exact Sciences 15.1, pp. 67-114.
Staal, Frits. 1995. "The Sanskrit of Science". In Journal of Indian Philosophy 23, pp. 73-127.
Subbarayappa, B. V. 1970. "India's Contributions to the History of Science". In Lokesh Chandra, et al., eds. India's Contribution to World Thought and Culture. Madras: Vivekananda Rock Memorial Committee, pp. 47-66.

The Ruins of Nalanda University
A Buddhist University, 5th Century
Bihar, India

Towards the Southeast of Patna is a village called the 'Bada Gaon', in the vicinity of which, are the world famous ruins of Nalanda University. Housing about 10,000 students and 2,000 teachers, this university attracted pupils from all over the world. A Walk in the ruins of the university, takes you to an era, that saw India leading in imparting knowledge, to the world - the era when India was a coveted place for studies. The university flourished during the 5th and 12th century.

The ruins of Nalanda university is spread over an area of 14 hectares. This university was totally built in Red clay bricks. The Nalanda university attracted scholars from all over the world. Even Chanakya or Kautilya was once a student of this university. This university was seat of knowledge for the world, the light of knowledge spread all over the world from Nalanda. Today only the memories of those glorious days are refreshed in the ruins. Whatever remains of the great university has been well preserved. Among the ruins one still recognizes the different sections of the place. Particularly the place of worship and the hostels are very distinct. The whole area is surrounded by beautiful lawns.
At the excavation site, Visitors going in for the monasteries and temples stand at the Eastern gate. The tourists going in for university ruins enter the site from the Western gate. The main temple area no. 3 is situated on the southern side of the site. The temple is surrounded by many small Stupas in a courtyard. The Vihar area no. 1 is the most important at this place. This Vihar has 9 storeys. The various levels are identified by the concrete courtyards and the walls & drains which are built one over the other. It is believed that the lower most Vihar was built by Devapal, the third king of the Pala dynasty.
This place saw the rise and fall of many empires and emperors who contributed in the development of Nalanda. Many monasteries and temples were built by them. King Harshwardhana gifted a 25m high copper statue of Buddha and Kumargupta endowed a college of fine arts here. Nagarjuna- a Mahayana philosopher, Dinnaga- founder of the school of logic and Dharmpala- the Brahmin scholar, taught here.
The famous Chinese traveller and scholar, Hieun-Tsang stayed here and has given a detailed description of the situations prevailing at that time. Careful excavation of the place has revealed many stupas, monasteries, hostels, stair cases, meditation halls, lecture halls and many other structures which speak of the splendour and grandeur this place enjoyed, when the place was a centre of serious study.
Hieun Tsang Memorial Hall:
Hieun Tsang was a Chinese traveller, who came to India in around 5th century. He has given a very detailed and vivid description of the Indian political and social conditions at that time. His writing is considered to be one of the most authentic sources of information of that period. Hieun Tsang was also attracted by the glory of Nalanda University. He came and stayed here, both as a student and as a teacher. As a student, he studied Yoga for six years under Acharya Shil Bhadra. He was in Nalanda for twelve years. The memorial Hall has been built in his memory.

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A fascinating historical account by
The Chinese pilgrim Hieun Tsang
(A student of Nalanda in 5th century)

The Royal Patrons of the University of Nalanda
Researched by Rev. H. Heras, S.J., M.A.
Journal of the Bihar and Orissa Research Society, PART I.
Vol. XIV 1928 pp. 1-23
Includes reference from I-Tsing, A Record of the Buddhist Religion
(Translated by Takakasu).

King Kumarra Gupta I is undoubtedly the founder of the university of Nalanda.
The Chinese pilgrim Hieun Tsang does not say that Kumara Gupta was a Buddhist, but says only that he "respected and esteemed" the law of Buddha and "honoured very highly" the Buddha, the dharma and the sangha. In fact he seems to have been a Vaienava. But such respect and esteem for Buddhism is not a strange thing in a Hindu monarch.

Moreover we cannot doubt that Kumara Gupta gave some endowments to the university, as some of the other kings mentioned by Hiuen Tsiang also did after him, so that the students being supplied with everything should not require to ask for anything. Thus the university could be called from the beginning of its existence Nalanda, i.e. "charity without intermission." As a matter of fact I-Tsing records the fact that the lands possessed by the university, that contained more than 200 villages, had been bestowed upon the institution "by kings of many generafions."
Of this king Hiuen Tsiang says:
"Buddhagupta-raja...... continued to labour at the excellent undertaking of his father. To the south of this he built another sangharama."
Accordingly Skanda Gupta continued the policy of his father towards the university. His patronage was specially shown in the fact that he built another sangharama to the south of that erected by his father. Such enlargement of the university was most likely carried out by Skanda Gupta after his victorious return from the west where he had crushed the power of the Hunas, then for the first time invading the plains of Aryavarta.

King Pura Gupta
Pura Gupta is said by Hiuen Tsiang to have "vigorously practised the former rules (of his ancestors), and he built east from this another sangharama." These words of the Chinese pilgrim point out two facts: first, the building of another college east of the one built by his brother; second, a more vigorous patronage policy in favour of the university, probably by granting privileges and endowments to the institution. We have already mentioned the fact of Pura Gupta's great devotion to Vasubandhu. It is not strange therefore that, either on his own accord or perhaps influenced by Vasubandhu, Pura Gupta should favour the institution even more than his deceased brother King Narasimha Gupta
Hitherto the famous university had not aparently suffered as yet any attack of any enemy. But by this time the Gupta Emperors had already lost their paramount sovereignty and had become feudatories of his foreign enemies the Hunas. The latter's king Mihirakula, whose capital seems to have been somewhere in Malwa, issued a decree during Narasimha Gupta's reign, by which he declared his purpose "to destroy all the (Buddhist) priests through the five Indies, to overthrow the law of Buddha, and leave nothing remaining."

The greatest number of Bhiksus undoubtedly resided in the kingdom of Narasimha Gupta. So this king, a fervent disciple of Vasubandhu, and who is said by the Chinese pilgrim to have "profoundly honoured the law of Buddha," as soon as news of the persecution begun by Mihirakula reached his ears "he strictly guarded the frontiers of his kingdom and refused to pay tribute.'' This was a declaration of war on the part of the Gupta sovereign. The Huna king accepted the challenge, entered the kingdom of Magadha and pursued Narasimha Gupta till the bay of Bengal.In the course of this campaign Mihirakula at the head of his army had to pass very near the university of Nalanda, for he first undoubtedly marched on Pataliputra, and only when he realised that the Gupta sovereign had fled towards the sea then he continued his march till the bay of Bengal.
This inroad of the Huna army was bound to be fatal to the kingdom of Magadha and specially to the Buddhist religion then protected and patronized by the Gupta monarchs. Mihirakula, beyond doubt, in his hatred of Buddhism destroyed all its buildings that he found in his way, and killed all its priests-- cruelties which he was shortly afterwards to repeat from his exile into Kashmir. Nalanda University was not far from the capital, Pataliputra, and its fame had also reached Mihirakula's ears. The buildings of Nalanda were then probably destroyed for the frst time, and its priests and students dispersed and perhaps kiiled.
But Mihirakula was finally defeated by the Gupta army and exiled to Kashmir by the victor.After this Narasimha Gupta, the great patron of Buddhism, could not permit that such an important institution of learning should perish. Hiuen Tsiang tells us that he built another sangharama (College) on the northeast side of the one built by his father.This sangharama (college) was still called "the college of Baladitya-raja" in the time of Hiuen Tsiang. Moreover he constructed a great vihara 300 feet high. "With respect to its magnificence," says Hiuen Tsiang, "its dimensions, and the statue of Buddha placed in it, it resembles the great vihara built under the Bodhi tree."
But besides the building of the sangharama and the vihara Nalanda undoubtedly owed to Narasimha Gupta the restoration of the whole university after the destruction by the Huna King. The new sangharama mentioned by Hiuen Tsiang was only an enlargement of the university; but the old buildings were partly reconstructed or newly built over the ruins of the former ones. This has been evidently proved in the course of the excavations. They have shown that some of the monasteries and other buildings have been erected on the ruins of earlier ones. Moreover in the time of Harsa-vardhana the main hall built by Kumara-Gupta I was still existing, either in its primitive form or partially reconstructed.This shows that after the destruction of the university the pristine plot was not abandoned. After this work of restoration was done and after the new sangharama and vihara were finished, Narasimha Gupla decided to commemorate the event with a great assembly. Hiuen Tsiang says that he "invited common folk and men of religion without distinction." The meeting of this assembly was a great succese; 10,000 priests flocked to Nalanda from every corner of India and even two monks came from far-off China.
Narasimha Gupta, on seeing the faith of these two foreign monks coming to the great celebrations at Nalanda from so distant a country, "was filled with gladness," says the biographer of Hiuen Tsiang. This gladness seems to have been the effect of a great spiritual consolation, for the Chinese pilgrim himself says that "the king then was affected by a profound faith." The result of this faith and gladness was that Narasimha Gupta resigned the crown and entered the sangha as a monk. We cannot doubt these two facts; Hiuen Tsiang records that "he gave up his country and became a recluse," while his biographer states even more explicitly that "he gave up his royal estate and became a recluse."
We know of only one episode of the life of Narasimha Gupta in the sangha. The Chinese pilgrim relates that "he (before being fully ordained) placed himself as the lowest of the priests, but his heart was always uneasy and ill at rest. 'Formerly (he said) I was a king, and the highest among the honourable; but now I have become a recluse, I am degraded to the bottom of the priesthood'." The poor ex-king, though living within the walls of his vihara, was still wishing to be the recipient of the wordly honours which he had been accustomed to in former days. He consequently manifested his grievance to the superiors of the sangha. It was consequently resolved, in order to please the royal disciple, that those monks who had not yet received the full orders should be classed not according to the number of years they had been lay disciples, but according to their natural years of life till the time of receiving full ordination. Narasimha seems to have been pleased with the decision. The only thing he could not stand was to be the last of the whole community. According to this change in the monastic customs, he had all the young monks behind --as he was then an old man--and his ambition was satisfied. "This sangharama,'' adds the Chinese pilgrim," is the only one in which this law exists." We do not know whether Narasimba Gupta was finally fully ordained.
Kumara Gupta II
After narrating Narasimha's life in the sangha, Hiuen Tsiang adds: "The king's son, called Vajra, came to the throne in succession." According to the Bhitari seal of Kumara Gupta II, the latter was the son and successor of Nsrasimha Gupta.It is a fact recorded by Hiuen Tsiang that "he again built on the west side of the convent a sangharama.
" A long succession of kings," says Hiuen Tsiang, "continued the work of building, using all the skill of the sculptor, till the whole is truly marvellous to behold." The appearance of the university after all these sangharamas and buildings had been constructed is said to have been "truly marvellous" by the Chinese pilgrim. Indeed his biographer writes a fine description of the university, like a bird's-eye view, which is worth quoting as showing what the university looked like during the first half of the seventh century A.D., after all those kings had embellished its monasteries and decorated its towers and observatories. Hwui Li's description is to the following effect:

"The richly adorned towers, and the fairy-like turrets, like pointed hill-tops, are congregated together. The observatories seem to be lost in the vapours of the morning, and the upper rooms tower above the clouds. From the windows one may see how the winds and the clouds produce new forms, and above the soaring eaves the conjunctions of the sun and moon may be observed. And then we may add how the deep translucent ponds bear on their surface the blue lotus, intermingled with the Kie-ni (Kanaka) flower of deep red colour and at intervals the Amra groves spread over all their shade. All the outside courts, in which are the priests' chambers, are of four stages. The stages have dragon projections and coloured eaves, the pearl-red pillars, carved and ornamented, the richly adorned balustrades, and the roofs covered with tiles that reflect the light in a thousand shades, these things add to the beauty of the scene."
During this period there occurred in Magadha several wars, which by the natural havoc consequent on any war, may also have been destructive of Nalanda. The above-mentioned Apshad Inscription of Adityasena mentions two defeats inflicted on king Isanavarman, the first by one Kumara Gupta (probably the same Kumara Gupta II) and the seoond by Damodara Gupta; while Isanavarman himself had previously defeated the Hunas. Then Mahasena Gupta won a victory over Susthivarman. The Haraha inscription of Isanavaraman also refers to the victories of this monarch over the lord of the Andhras "who had thousands of threefold rutting elephants," over the Sulikas "who had an army of countless galloping horses, " and over the Gaudas "living on the seashore." A partial destruction of Nalanda caused by these wars (some of which were evidently fought in the territory of Magadha) may well have taken place.
Harsha-vardhana Vajra, i.e. Kumara Gupta II, seems to be the last king of the Gupta family mentioned by Hiuen Tsiang in connection with the university of Nalanda. Moreover not even other kings of the same kingdom of Magadha, but belonging to other dynasties, are referred to by the Chinese pilgrim. But he adds the name of a king of another kingdom to this list of patrons of the university of Nalanda. After having mentioned Vajra and his doings, he says: "After this a king of Central India."
This king of Central India, that appears after the extinction of the Gupta family before the arrival of Hiuen Tsiang in India, cannot be other than Harsa-vardhana of Kanauj. The same Hiuen Tsiang refer clearly to him in other two passages of his account in connection with the university. That this monarch had positively and openly declared leanings towards Buddhism is clear from other passages of Hiuen Tsiang's travels. When we read for instance Hiuen Tsiang's account of the assembly of Kanauj convoked by Harsa for propagating the doctrines of Mahayana, we cannot doubt that the great emperor had accepted in his heart the faith of Buddha.

The first dealings of Harsa with Nalanda seem, so it appears, to be connected with a double tragedy of his family. His sister Rajyasri had been married to the Maukhari king Grahavarman.This king, some years later, had been defeated and killed by king Deva Gupta of Malwa and after his death Rajyasri had been cast into prison by the victor. Harsa's brother, Rajya-vardhana, then the king at Thanesar, could not stand this affront on his family, marched against Deva Gupta and defeated him. But it so happened just at this moment that Sasanka, king of Gauda in Eastern Bengal, entered Magadha as a friend of Rajya-vardhana, but in secret alliance with the Malwa king. Accordingly Sasanka treacherously murdered Rajya-vardhana. It was most likely on this occasion that he destroyed the sacred places of Buddhism, as related by Hiuen Tsiang: " Lately Sasanka-raja" says he,''when he was overthrowing and destroying the law of Buddha, forthwith came to the place where that stone is, for the purpose of destroying the sacred marks (Buddha's foot-prints). Having broken it into pieces, it came whole again, and the ornamental figures as before; then he flung it into the river Ganges."
"In later times," the same Hiuen Tsiang goes on to say, '' Sasanka-raja, being a believer in heresy, slandered the religion of Buddha and through envy destroyed the convents and cut down the Bodhi tree (at Buddha Gaya), digging it up to the very springs of the earth; but yet he did not get to the bottom of the roots. Then he burnt it with fire and sprinkled it with the juice of sugar-cane, desiring to destroy them entirely, and not leave a trace of it behind." Such was Sasanka's hatred towards Buddhism.
Hence we cannot imagine this king going from the Ganges to Gaya and passing so near Nalanda, the greatest centre of Buddhism in those days, without leaving there the effects of his bigotry. That most likely was a new occasion on which the buildings of Nalanda were razed to the ground and its inhabitants murdered or dispersed. On hearing of the murder of his brother, Harsa resolved at once to march against the treacherous king of Gauda, and both the Harsa Charita and Hiuen Tsiang agreed as to the colossal success of Harsa's efforts. After having driven Sasanka to Bengal we cannot doubt that Harsa, the enthusiastic disciple of
Mahayana Buddhism, restored the university of Nalanda to its pristine grandeur, just as Purnavarma repaired the damages caused by Sasanka at Buddh Gaya.

But this was not all. Harsa, called by Hiuen Tsiang "a king of Central India," "built to the north of this a great sangharama." The Chinese pilgrim seems to indicate that the sangharama built by Harsa was greater than those built by other kings in the precincts of the university, for this is the only one called ''great" by him. Hiuen Tsiang mentions another building due also to the devotion and munificence of Harsa. "To the south of this," says he, " is a vihara of brass built by Siladitya-raja." It is well known that Siladitya-raja is the name given to Harsa by the Chinese pilgrim, a title which is also confirmed by numis- matics.This vihara was still under construction at the time of Hiuen Tsiang's stay at the university.
"Although it is not yet finished," he adds, "yet its intended measurement, when finished, will be hundred feet." But Hiuen Tsiang's biographer, who wrote some years later, seems to have received some more information about this building after its completion. In fact Hwui Li says that "it was renowned through all countries." The vihara, according to Hwui Li's information, was not made all of brass, but only " covered with brass plates." Indeed the appearance of the building was "magnificent and admirable." In fact the Hinayana monks of Orissa envied the Mahayana monks of Nalanda so rich and gorgeous a building."
Moreover in the time of Hiuen Tsiang Harsa had the purpose of dedicating an image of Buddha " in the hall of the monarch who first began the sangharama." This seems to be an allusion to the first sangharama built by Kumara Gupta I.Finally Harsa's patronage is also shown by the numerous endowments be granted to the university. " The king of the country," says Hwui Li, "respects and honours the priests, and has remitted the revenues of about 100 villages for the endowment of the convent. Two hundred householders in these villages, day by day, contribute several piculs of ordinary rice, several hundred catties in weight of butter and milk." The biographer here draws a consequence that discloses the great importance of these endowments of Harsa. " Hence the students here, being so abundantly supplied, do not require to ask for the four requisites (clothing, food, bedding and medicine). This is the source of the perfection of their studies, to which they have arrived." Hiuen Tsiang himself also informs us that when Harsa decided to erect an image of Buddha in the singharama of Kumara Gupta, he said too: "I will feed forty priests of the congregation every day to show my gratitude to the founder."
These endowments and grants of Harsa were most likely confirmed by official documents adorned with his seal. In fact two seals of Harsa have been found in Nalanda in the course of the excavations. All these favours and donations of the great emperor were crowned by the construction of a lofty wall enclosing all the buildings of the university.His intention seems to have been to defend the institution of any other possible hostile inroad.
Kings of other Countries
The Chinese pilgrim speaking of a brick vihara of Nalanda, where an image of Tara Bodhisattva was venerated, says as follows: --" The kings and ministers and great people of the neighbouring countries offer exquisite perfumes and flowers, holding gem-covered flags and canopies, whilst instruments of metal and stone resound in turns, mingled with the harmony of flutes and harps. These religious assemblies last for seven days."
Who were these kings of the neighbouring countries in the time of Hiuen Tsiang besides the great Vardhana? Unfortunately the Chinese pilgrim does not give any clue for ascertaining this doubt. Anyhow six were the main kingdoms round Harsa's empire: the kingdom of the Maukharis, the kingdom of Gauda in Bengal, the kingdom of Kamarupa in Assam, the kingdom of Nepal, the kingdom of the Valabhis in Saurastra and tile kingdom of the Chalukyas in the Deccan. Let us examine separately the possibility of the patronage of Nalanda by the kings of these countries.

(a) The Maukharis.--Some of the Maukharis may undoubtedly be counted among the patrons of Nalanda. Two of their seals have also been found at Nalanda next to the seals of Harsa.Moreover Purnavarma, whom I consider to be the last Maukhari, seems to have had great affection for Buddha and his doctrines. Hiuen Tsiang tells us that when hearing of the destruction caused to the Bodhi tree by the fanaticism of Sasanka, Purnavarma exclaimed: " The sun of wisdom having set, nothing is left but the tree of Buddha, and this they now have destroyed; what source of spiritual life is there now." "He then," continues Hiuen Tsiang, "cast his body on the ground overcome with pity; then with the milk of a thousand cows he again bathed the roots of the tree, and in a night it once more revived and grew to the height of some 10 feet. Fearing lest it should be again cut down, he surrounded it with a wall of stone 24 feet high.'' Such a great devotion for the law of Buddha surely compelled also Purnavarma to patronize the Nalanda University, specially after its destruction by the same Sasanka who uprooted the Bodhi tree. In fact the same Hiuen Tsiang mentions a "pavilion of six stages" made at Nalanda by Purnavarma to enshrine a copper statue of Buddha 80 feet high.
(b) Gauda in Bengal.--We have seen that its king Sasanka was a declared enemy of Buddhism. His relations with Nalanda seem to have been purely negative and destructive.
(c) Kamarupa in Assam.--The king of Kamarupa contemporary of Harsa was named Bhaskaravarman. He was a Brahmana by caste and by faith, but he respected and was much interested in the law of Buddha. When he came to know of the existence of a Chinese pilgrim, Hiuen Tsiang, at the Nalanda University he sent him three different messages inviting him to his court, till his wish was satisfied. Later on we see him accompanying Harsa-vardhana in the great Buddhist assembly of Kanauj, where Mahayana Buddhism was propounded.He must undoubtedly be counted among the patrons of the Nalanda University. One of his seals, found at Nalanda next to those of Harsa, seems to prove the same fact.
(d) Nepal.--Hiuen Tsiang gives some information about this country and its king. His name was Amsuvarman, the founder of the Thakuri dynasty. He "was a descendant of the Licchavis. The Chinese pilgrim refers to his intellectual abilities and to his religion. As regards the former he says that he was distinguished for his learning and ingenuity. He himself had composed a work on 'sounds'; he esteemed learning and respected virtue, and his reputation was spread everywhere" As to his religion Hiuen Tsiang says as follows: "His mind is well informed, and he is pure and dignified in character. He has a sincere faith in the law of Buddha.'' In fact one of the inscriptions of this king, published by Pandit Bhagvanlal Indraji, shows on the top the wheel of the law, between two deer, that is a symbol of the first sermon of Buddha at the Deer Park, Sarnath.The literary likings of this king and his religious faith make quite probable that he himself patronized in some way or other the university of Nalanda, specially if we consider that he paid homage to Harsa-vardhana, as the introduction of Sriharsa era clearly shows, and that he visited Harsa's kingdom, a fact recorded in the Parvaviya Vamsavali.
(e) The Valabhis of Saurastra.-According to Hiuen Tsiang the contemporary Valabhi king was Dhruvapata. He seems to be king Siladitya VI, who is also surnamed Dhrubhata or Dhruvabhata, i.e. "the constant warrior." About his religion the Chinese pilgrim says "Quite recently he has attached himself sincerely to faith in the three 'precious ones' (Buddha. dharma and sangha)." He moreover describes his character and likings as follows: " He is of a lively and hasty disposition, his wisdom and statecraft are shallow. He esteems virtue and honours the good; he reverences those who are noted for their wisdom. The great priests who come from distant regions he practically honours and respects." This seems to give some probability to his being one of the benefactors of Nalanda University.
(f) The Chalukyas of the Deccan.--The contemporary sovereign of the Deccan was Pulakesin II, the greatest monarch of the Chalukyan dynasty. Hiuen Tsiang says that "'his beneficent actions are felt over a great distance." Nevertheless we are not aware of his leanings towards Buddhism. Moreover, himself being an enemy of Harsa, whom ha defeated near the Narbada, it is not probable that he would favour the Nalanda University within the boundaries of his enemy's dominions. Besides these sovereigns there were in northern India several petty rajas who had acknowledged the sovereignity of Harsa. They also perhaps favoured at times the university of Nalanda. Hiuen Tsiang says that there were twenty of these kings round Harsa at the Charity Assembly he witnessed at Prayaga (Allahabad).
A Plan of the University of Nalanda
As a complement of our study about the royal patrons of Nalanda,a probable plan of the university at the time of Hiuen Tsiang's visit, and according to the data furnished by him, will not, I think, be out of place. Certainly this plan cannot be without errors. The information is not great and scattered here and there without giving the distances between buildings and buildings, excepting in two or three cases. Anyhow this rough sketch will give a general idea of what that famous institution was like and will show the munificence of its royal patrons and benefactors.
1. Sangharama or college built by Kumara Gupta I "on
a lucky spot." (Beal, o. c., p. 168.)
2. To the south of this, sangharama built by Skanda Gupta. (Ibid.)
3. To the east of this, sangharama built by Purra Gupta. (Ibid.)
4. On the north-east side, sangharama built by Narasimha Gupta. (Ibid).
5. On the west side of the convent, I understand, of the first original monastery, sangharama built by Kumara Gupta II (p. 170). The Life of Hiuem Tsiang, p.111, disagrees. Hwui Li says only that this sangharama was "to the north. " This uncertainty and the fact that he never saw Nalanda causes me to prefer Hiuen Tsiang's statement.
6. To the north of this, great sangharama built by Harsa--vardhana (Beal, o. c., p. 170.) The Life of Hiuen Tsiang, l.c., says that this sangharama was built " by the side'' of the one built by Kumara Gupta II.
7. " On the western side of the sangharama, at no great distance, is a vihara." (Beal, o. c, p. 172.)
8. "To the south 100 paces or so is a small stupa." (Ibid.)
9. "On this southern side is a standing figure of Avalokitesvara Bodhisattva.'' (Ibid.)
10. "To the south of this statue is a stupa in which are remains of Buddha's hair and nails." (Ibid., 173.)
11. "To the west of this, outside the wall, and by the side of a tank, is a stupa." (Ibid.) p. 22
12. Tank. (Vide No. 11.)
13. "To the south-east about 50 paces, within the walls, is an extraordinary tree, about eight or nine feet in height, of which the trunk is twofold, " (Ibid.)
14. "Next to the east there is a great vihara about 200 feet in height." (Ibid.) In the plan I placed this vihara much towards the east; otherwise there is no room for the following viharas north of this. According to this arrangement, the vihara No. 16, built by Narasimha Gupta, comes in the neighbourhood of the sangharama built by the same monarch, a fact that does not look improbable.
15. "After this, to the north 100 paces or so, is a vihara in which is a figure of Avalokitesvara Boddhisattva. " (Ibid).
16. "To the north of this vihara is a great vihara, in height about 300 feet, which was built by Baladitya- raja." (Natrasimha Gupta) (Ibid.)
17. "To the north-east of this is a stupa. " (Ibid. p. 174.)
18. "To the north-west is a place where the four past Buddhas sat down." (Ibid.)
19. "To the south of this is a vihara of brass built by Siladitya-raja." (Harsa-vardhana) (Ibid.) Accordingly this vihara built by Harsa is not far from the great sangharama also built by him.
20. "Next to the eastward two hundred paces or so, outside the walls, is a figure of Buddha standing uprigh and made of copper. Its height is about 80 feet. A pavilion of six stages is required to cover it." (Ibid.) This is the pavilion built by Purnavarma.
21. "To the north of this statue two or three li, in a vihara constructed of brick, is a figure of Tara Bodhisattva. This figure is of great height and its spiritual appearance very striking." (Ibid.) p. 23
22. "Within the souhern gate of the wall is a large well," (Ibid, p. 175.) The phrase of the pilgrim seems to show that this well was close to the gate. I had no special reason for putting it on the east side of the same.
23. "High wall" built round these edifices by Harsa-vardhana. (Ibid., p, 179.)
24. The only gate to enter the premises of the university. (Ibid.) Further on Hiuen Tsiaug says that this gate was " southern." (Ibid., p. 175.) The following quotation discloses the importance of this gate. "If men of other quarters desire to enter and
take part in the discussions, the keeper of the gate proposes some hard questions; many are unable to answer, and retire. One must have studied deeply both old and new books before getting admission. Those students, therefore, who come here as strangers, have to show their ability by hard discussion, those who failed compared with those who succeed are as seven or eight to ten." (Ibid., p. 171.)
The precise extension of the area covered by the buildings of the university is not said by Hiuen Tsiang. Anyhow Hwui Li states that this sangharama (the whole university) " is the most remarkable for grandeur and height " all over India.(1) He also states that "the priests belonging to the convent, or strangers residing therein always reach to the number of 10,000."(2) Finally I-Tsing, who travelled through India towards the close of the same century, informs us: There are eight halls and three hundred apartments in this monastery."(3) These scanty data will give some idea of the great extension of the university of Nalanda.

Nalanda - An ancient University
a travelogue by Surajit Basu
A modern map of the ancient university shows us the chaityas and the monasteries, where the students learnt and lived. The chaityas were the temples, centres of meditation and learning while the monasteries were the hostels.
2,000 teachers and 10,000 students stayed and studied in this university. Some came from other countries, other cultures. Like Hiuen Tsang from China who wrote about his days here. In the 7th century. He left an elaborate description of the excellence of Nalanda.
Nalanda - An ancient University
A travelogue by Surajit Basu
Welcome to Nalanda, the ancient university. Set in India,
close to Gaya in Bihar, this site was lost for hundreds of
years and rediscovered.

How to get there : The nearest airport is Patna, 90 km away.
Train services to Nalanda are available. Trains to Rajgir,
10 km away, are also available.

Om Tat Sat

(My humble salutations to  Swamy Sri Sadasiva Teertha and professors    for the collection)


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