Islam's Other Victims: India

Adapted from The Sword of the Prophet: A Politically-Incorrect Guide to Islam by Dr. Serge Trifkovic.

The fundamental leftist and anti-American claim about our ongoing conflict with political Islam is this: whatever has happened or does happen, it’s our fault. We provoked them into it by being dirty Yankee imperialists and by unkindly refusing to allow them to destroy Israel. But two things make crystal clear that this is not so:

1. The political arm of Islam has been waging terroristic holy war on the rest of the world for centuries.

2. It has waged this war against civilizations that have nothing to do with the West, let alone America.

This is why the case of Moslem aggression against India proves so much. Let’s look at the historical record.

India prior to the Moslem invasions was one of the world’s great civilizations. Tenth century Hindustan matched its contemporaries in the East and the West in the realms of philosophy, mathematics, and natural science. Indian mathematicians discovered the number zero (not to mention other things, like algebra, that were later transmitted to a Moslem world which mistaken has received credit for them.) Medieval India, before the Moslem invasion, was a richly imaginative culture, one of the half-dozen most advanced civilizations of all time. Its sculptures were vigorous and sensual, its architecture ornate and spellbinding. And these were indigenous achievements and not, as in the case of many of the more celebrated high-points of Moslem culture, relics of pre-Moslem civilizations that Moslems had overrun.

Moslem invaders began entering India in the early 8th century, on the orders of Hajjaj, the governor of what is now Iraq. (Sound familiar?) Starting in 712 the raiders, commanded by Muhammad Qasim, demolished temples, shattered sculptures, plundered palaces, killed vast numbers of men — it took three whole days to slaughter the inhabitants of the city of Debal — and carried off their women and children to slavery, some of it sexual. After the initial wave of violence, however, Qasim tried to establish law and order in the newly-conquered lands, and to that end he even allowed a degree of religious tolerance. but upon hearing of such humane practices, his superior Hajjaj, objected:

"It appears from your letter that all the rules made by you for the comfort and convenience of your men are strictly in accordance with religious law. But the way of granting pardon prescribed by the law is different from the one adopted by you, for you go on giving pardon to everybody, high or low, without any discretion between a friend and a foe. The great God says in the Koran [47.4]: "0 True believers, when you encounter the unbelievers, strike off their heads." The above command of the Great God is a great command and must be respected and followed. You should not be so fond of showing mercy, as to nullify the virtue of the act. Henceforth grant pardon to no one of the enemy and spare none of them, or else all will consider you a weak-minded man."

In a subsequent communication, Hajjaj reiterated that all able-bodied men were to be killed, and that their underage sons and daughters were to be imprisoned and retained as hostages. Qasim obeyed, and on his arrival at the town of Brahminabad massacred between 6,000 and 16,000 men.

The significance of these events lies not just in the horrible numbers involved, but in the fact that the perpetrators of these massacres were not military thugs disobeying the ethical teachings of their religion, as the European crusaders in the Holy Land were, but were actually doing precisely what their religion taught. (And one may note that Christianity has grown up and no longer preaches crusades. Islam has not. As has been well-documented, jihad has been preached from the official centers of Islam, not just the lunatic fringe.)

Qasim’s early exploits were continued in the early eleventh century, when Mahmud of Ghazni, "passed through India like a whirlwind, destroying, pillaging, and massacring," zealously following the Koranic injunction to kill idolaters, whom he had vowed to chastise every year of his life.

In the course of seventeen invasions, in the words of Alberuni, the scholar brought by Mahmud to India,

"Mahmud utterly ruined the prosperity of the country, and performed there wonderful exploits, by which the Hindus became like atoms of dust scattered in all directions, and like a tale of old in the mouth of the people. Their scattered remains cherish, of course, the most inveterate aversion toward all Moslems."

Does one wonder why? To this day, the citizens of Bombay and New Delhi, Calcutta and Bangalore, live in fear of a politically-unstable and nuclear-armed Pakistan that unlike India (but like every other Moslem country) has not managed to maintain democracy since independence.

"In the middle of the city there was a temple larger and finer than the rest, which can neither be described nor painted." The Sultan [Mahmud] was of the opinion that 200 years would have been required to build it. The idols included "five of red gold, each five yards high," with eyes formed of priceless jewels. "The Sultan gave orders that all the temples should be burnt with naphtha and fire, and leveled with the ground."

In the aftermath of the invasion, in the ancient cities of Varanasi, Mathura, Ujjain, Maheshwar, Jwalamukhi, and Dwarka, not one temple survived whole and intact. This is the equivalent of an army marching into Paris and Rome, Florence and Oxford, and razing their architectural treasures to the ground. It is an act beyond nihilism; it is outright negativism, a hatred of what is cultured and civilized.

In his book The Story of Civilization, famous historian Will Durant lamented the results of what he termed "probably the bloodiest story in history." He called it "a discouraging tale, for its evident moral is that civilization is a precious good, whose delicate complex order and freedom can at any moment be overthrown by barbarians invading from without and multiplying from within."

Moslem invaders "broke and burned everything beautiful they came across in Hindustan," displaying, as an Indian commentator put it, the resentment of the less developed warriors who felt intimidated in the encounter with "a more refined culture." The Moslem Sultans built mosques at the sites of torn down temples, and many Hindus were sold into slavery. As far as they were concerned, Hindus were kafirs, heathens, par excellence. They, and to a lesser extent the peaceful Buddhists, were, unlike Christians and Jews, not "of the book" but at the receiving end of Muhammad’s injunction against pagans: "Kill those who join other gods with God wherever you may find them." (Not that being "of the book" has much helped Jewish and Christian victims of other Moslem aggressions, but that’s another article.)

The mountainous northwestern approaches to India are to this day called the Hindu Kush, "the Slaughter of the Hindu," a reminder of the days when Hindu slaves from Indian subcontinent died in harsh Afghan mountains while being transported to Moslem courts of Central Asia. The slaughter in Somnath, the site of a celebrated Hindu temple, where 50,000 Hindus were slain on Mahmud’s orders, set the tone for centuries.

The gentle Buddhists were the next to be subjected to mass slaughter in 1193, when Muhammad Khilji also burned their famous library. By the end of the 12th century, following the Moslem conquest of their stronghold in Bihar, they were no longer a significant presence in India. The survivors retreated into Nepal and Tibet, or escaped to the south of the Subcontinent. The remnants of their culture lingered on even as far west as Turkestan. Left to the tender mercies of Moslem conquerors and their heirs they were systematically destroyed, sometimes—as was the case with the four giant statues of Buddha in Afghanistan in March 2001—up to the present day.

That cultivated disposition and developed sensibility can go hand in hand with bigotry and cruelty is evidenced by the example of Firuz Shah, who became the ruler of northern India in 1351. This educated yet tyrannical Moslem ruler of northern India once surprised a village where a Hindu religious festival was celebrated, and ordered all present to be slain. He proudly related that, upon completing the slaughter, he destroyed the temples and in their place built mosques.

The Mogul emperor Akbar is remembered as tolerant, at least by the standards of Moslems in India: only one major massacre was recorded during his long reign (1542-1605), when he ordered that about 30,000 captured Rajput Hindus be slain on February 24, 1568, after the battle for Chitod. But Akbar’s acceptance of other religions and toleration of their public worship, his abolition of poll-tax on non-Moslems, and his interest in other faiths were not a reflection of his Moslem spirit of tolerance. Quite the contrary, they indicated a propensity for free-thinking in the realm of religion that finally led him to complete apostasy. Its high points were the formal declaration of his own infallibility in all matters of religious doctrine, his promulgation of a new creed, and his adoption of Hindu and Zoroastrian festivals and practices. This is a pattern one sees again and again in Moslem history, down to the present day: whenever one finds a reasonable, enlightened, tolerant Moslem, upon closer examination this turns out to be someone who started out as a Moslem but then progressively wandered away from the orthodox faith. That is to say: the best Moslems are generally the least Moslem (a pattern which does not seem to be the case with other religions.)

Things were back to normal under Shah Jahan (1593-1666), the fifth Mogul Emperor and a grandson of Akbar the Great. Most Westerners remember him as the builder of the Taj Mahal and have no idea that he was a cruel warmonger who initiated forty-eight military campaigns against non-Moslems in less than thirty years. Taking his cue from his Ottoman co-religionists, on coming to the throne in 1628 he killed all his male relations except one who escaped to Persia. Shah Jahan had 5,000 concubines in his harem, but nevertheless indulged in incestuous sex with his daughters Chamani and Jahanara. During his reign in Benares alone 76 Hindu temples were destroyed, as well as Christian churches at Agra and Lahore. At the end of the siege of Hugh, a Portuguese enclave near Calcutta, that lasted three months, he had ten thousand inhabitants "blown up with powder, drowned in water or burnt by fire." Four thousand were taken captive to Agra where they were offered Islam or death. Most refused and were killed, except for the younger women, who went into harems.

These massacres perpetrated by Moslems in India are unparalleled in history. In sheer numbers, they are bigger than the Jewish Holocaust, the Soviet Terror, the Japanese massacres of the Chinese during WWII, Mao’s devastations of the Chinese peasantry, the massacres of the Armenians by the Turks, or any of the other famous crimes against humanity of the 20th Century. But sadly, they are almost unknown outside India.

There are several reasons for this. In the days when they ruled India, the British, pursuing a policy of divide-and-rule, whitewashed the record of the Moslems so that they could set them up as a counterbalance to the more numerous Hindus. During the struggle for independence, Gandhi and Nehru downplayed historic Moslem atrocities so that they could pretend a facade of Hindu-Moslem unity against the British. (Naturally, this façade dissolved immediately after independence and several million people were killed in the religious violence attendant on splitting British India into India and Pakistan.) After independence, Marxist Indian writers, blinkered by ideology, suppressed the truth about the Moslem record because it did not fit into the Marxist theory of history. Nowadays, the Indian equivalent of political correctness downplays Moslem misdeeds because Moslems are an "oppressed minority" in majority-Hindu India. And Indian leftist intellectuals always blame India first and hate their own Hindu civilization, just their equivalents at Berkeley blame America and the West.

Unlike Germany, which has apologized to its Jewish and Eastern European victims, and Japan, which has at least behaved itself since WWII, and even America, which has gone into paroxysms of guilt over what it did to the infinitely smaller numbers of Red Indians, the Moslem aggressors against India and their successors have not even stopped trying to finish the job they started. To this day, militant Islam sees India as "unfinished business" and it remains high on the agenda of oil-rich Moslem countries such as Saudi Arabia, which are spending millions every year trying to convert Hindus to Islam.

One may take some small satisfaction in the fact that they find it rather slow going.

Serge Trifkovic received his PhD from the University of Southampton in England and pursued postdoctoral research at the Hoover Institution at Stanford. His past journalistic outlets have included the BBC World Service, the Voice of America, CNN International, MSNBC, U.S. News & World Report, The Washington Times, the Philadelphia Inquirer, The Times of London, and the Cleveland Plain Dealer. He is foreign affairs editor of Chronicles: A Magazine of American Culture. This article was adapted for Front Page Magazine by Robert Locke.

History of 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, Syriac, 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, traveling 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 become 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.

Notes:

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).

References:

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: India 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

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