1. Most of the great discoveries and inventions of Europe is so proud would have been impossible without a developed system of mathematics, and this in turn would have been impossible if Europe had been shackled by the unwieldy system of Roman numerals. The unknown man who devised the new system was from the world's point of view the most important son of India. His achievement, though easily taken for granted, was the work of an analytical mind of the first order, and he deserves much more honor than he has so far received. (A. L. Basham, " The Wonder that was India", P- 498, pub- 1989, Rupa &co. New Delhi).

2. The Indian system of counting is probably the most successful intellectual innovation ever devised by human beings. It has been universally adopted. It is the nearest thing we have to a universal language. When the Chinese encountered the Indian system in the eight century, they adopted the Indian circular zero symbol and a full-place value notation with nine numerals, they adopted the Indian circular zero symbol and a full-place value notation with nine numerals. (John D. Barrow 's book " The Book of Nothing" Pages 35-52)

3. The earliest example of the use of the Indian zero is in AD458 when it appeared in a surviving Jain work on cosmology, but indirect evidence indicates that it must have been in use as early as 200 BC. At first, it was denoted by a dot, later, the familiar circular symbol, 0, replaced the dot. Indian decimal system was a regular one, with each level ten times the previous one, zero also acted as an operator. Thus, adding a zero to the end of a number string effected multiplication by 10 just as it does for us. (From John D. Barrow 's book " The book of Nothing" pages- 35-52)

4. In AD 628, the Indian astronomer Brhamagupta defined zero in this way and spelled out the algebraic rules for adding, subtracting, multiplying and most strikingly of all dividing with it. For example, " When sunya is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by sunya becomes sunya." (From John D. Barrow's book " The book of Nothing" Pages-35-52)

5. We have seen that our numerical zero derives originally from the Hindu sunya, meaning void or emptiness, deriving from the Sanskrit name for the mark denoting emptiness or sunya-bindu, meaning an empty dot. These developed between the sixth and eight centuries. By the ninth century the assimilation of Indian mathematics by Arab world led to the literal translation of sunya into Arabic as as-sifr, which also means empty or the 'absence of anything". The Arab world sifr was first transcribed into Medieval Latin in the thirteenth century in the two forms cifra or zefirum and into Greek as tau as an abbreviation for zero. In the fourteenth-century Italian, this second form changed to zefiro or zefro or zevero, which was eventually shortened in the Venetian dialect to zero, which we still use in English and French. (From John D. Barrow 's book " The Book of Nothing" Pages 35-52)

6. No description of ancient Indian mathematics can be complete without reference to the Shulva Sutra, which belongs to the literature of Vedic times (c.1500-c.200), Of the six parts of the Vedas (Sad-Vedangas), the sixth Vedanga Kalpa. The name itself means rules of measurement.It is interesting that since lengths were measured by ropes , the word shulva later came to be known as rope. The origins of the sutras can be traced to the Vedas and they may have been known at least eight to nine centuries B.C.The yajnas (sacrifices) were performed in Aryan/Vedic times to propitiate the divine powers or more generally as parts of religious rites. The size of the platform for yajnas and other related issues provoked questions of measurement and hence of geometry. Shulva Sutra contains, for example, Pythagoras' theorem but not the proof of the theorem, as Euclid's Elements does. Nevertheless, as a correct result, the statement should be renamed as the Shulva theorem.
   ( Jayant V. Narlikar, " The Scientific Edge" Page 4-5, pub-2003, penguin books , New Delhi)

7. Indian astronomer Brhamagupta defines infinity as the number that results from dividing any other number by zero and sets up a general system of rules for multiplying and dividing positive and negative quantities. (From John D. Barrow's book " The book of Nothing")

8. Earlier mathematicians had taught that x/0=x , Bhaskara proved that it was infinity . He also established mathematically, what has been recognized in Indian theology at least a millennium earlier, that infinity, however divided, remains infinite, represented by the equation ( A. L. Basham " The wonder that was India").

9. Good approximations of irrational numbers like pie and ^2 were known to ancient Indian mathematicians. Again we find these expressions given without proof. For example, the shulva sutra gives the following approximation for ^2 but without explicit proof:
   1+1/3+1(3x4) - 1/(3x4x34)
   For pie, Aryabhat gave the modern approximate value of 3.1416, expressed in the form of a fraction 62832/ 20000. Later Indian mathematicians improved the value of pie, much more than accurate than that of the Greeks, to nine places of decimals. (" The scientific Edge" and " The wonder that was India") The Baudhayana and the Apastambha sutras belonging to the Krishna Yajurveda describe indeterminate equations of the first degree, more commonly known as Diophantine equations because of their Greek origin. The Greek discovery, however, came much later, and it is more appropriate to recognize their origin as coming from the Shulva Sutra.

10. .In 1881, an unexpected find was unearthed in the village of Bakshali about seventy kilometers from the archeological site of Takshashila near Peshawar. This is a seventy page manuscript written on bhoorjapatras (birch bark) in the shrada script and in the gatha dilect of prakrit, which was prevalent in that part of India during the reign of the Kushnas and dates it to around 200 B.C. The manuscript contains mathematical results of high order including quadratic equations, finding square roots of numbers that are not perfect squares and arith matic geometric progressions. The Bakshali script therefore gives a fair idea of the advanced level of arithmetic and algebra in India of two millennia ago. ( J. V. Narlikar, "The Scientific Edge")

11. The precession of the equinoxes was known, and calculated with some accuracy by medieval astronomers, as were the lengths of the year, the lunar month, and other astronomical constants. These calculations were reliable for most practical purposes, and many cases more exact than those of the Greco-roman world. Eclipses were forecast with accuracy and their true cause understood.( A L Basham, " The wonder that was India")

Anulomagatirnausthah pashyatyachalam vilomagam yadvat

Achalani bhani tadvat sampashchimgani lankayam

- aryabhatiya 4.9

In this sloka he gives the analogy of a person going on a boat who sees fixed objects on the land going in the direction opposite to his, and he argues that the fixed stars likewise appear to go westwards because they are viewed from the moving surface of the earth. Here, Aryabhat is pointing to the spin of the earth around its axis from west to east, which gives rise to the apparent motion of the stars in the reverse direction. The analogy is exact and clear. Yet his contemporaries ignored this statement from a respected teacher and scholar like Aryabhat. It is creditable and rewarding to be slightly ahead of your contemporaries, it is much more creditable but not at all rewarding to be way ahead of them. For them they do not understand what you are saying and may ridicule your ideas. This happened to Aryabhata, too. The prevailing geocentric view did not allow one to think of the alternative of a spinning Earth. So Aryabhata's ideas remained buried and were long forgotten by the time the heliocentric view of Nicolous Copernicus (1473-1543). ( J. V. Narlikar, "The Scientific Edge").

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Bhaskara was a mathematician of the twelfth century who was both a mathematician and astronomer. One example of Bhaskara's work will suffice to illustrate the depth of his mathematical ability.In 1657, the famous French mathematician Pierre de Fermat sent a problem to his friend Bernard Frencile de Bessy. The problem asked for solutions of the equation

where X and Y are integers. They could not solve the problem, and it was only in 1732 that another great mathematician, Leonhard Euler, solved it. However it is now realized that the problem had already been solved in 1150 by Bhaskara II who gave the smallest such numbers as X = 22,61, 53,980 and y = 1,76,63,19,049. The method is called Chakravala method and is given in Bhaskara's Sidddhanta Shiromoni.But any account of Bhaskara's mathematical work will be incomplete without mentioning his book of mathematical problems, Lilavati, which was supposedly addressed to his daughter of the same name. Lilavati presents the reader with attractive problems poetically described and relating to contemporary life. Consider the following example:

The square root of half the total number of a swarm of bees went to a malati tree, followed by another eight ninth of the total. One bee was trapped inside the lotus flower, while his mate came humming in response to his call O lady, tell me how many bees were there in all?

This problem can be solved algebraically by using quadratic equation, the answer being that there were seventy-two bees in all. Bhaskara was interested in applying mathematical techniques to astronomy, and his work as presented in the Grahaganitam and the Goladhyaya are the culmination of a series of works in spherical astronomy by preceding astronomers like Aryabhata, Brahmagupta, Mahavira and others. He seems to have been close to developing the idea of calculus, writing formulae (similar to d sin x = cos x dx) that follow from differential calculus. He also seems aware that the derivate vanishes at the maxima or minima of the function, again a concept of modern differential calculus.

( J V Narlikar, The Scientific Edge)

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• A scientific approach to understanding the functions of human body and treating it accordingly also started in the Vedic era. As it was a part of the Vedic knowledge, the study acquired the name Ayurveda ( sacred knowledge of life).The ancient knowledge is today available principally from four basic texts, the Charaka-samhita, the Sushruta-samhita, the Ashtanga-Hridaya and the Astanga-sangraha, the most famous of these being the first one.

• Charaka is not the name of a single person, but it may be the name of the tribe mentioned in the KRISHNA YAJURVEDA. The main body of the text of the charaka-samhita was written around the seventh century B.C., around the time just preceding Buddha, although it would have contained knowledge known and in practice much earlier. It divides the medical studies into eight parts: surgery with implements, minor operations performed with superficial pricks, medical treatment of bodily ailment, ghostly treatment, medicines relating to diseases of women and children, treatments of venom and bites, chemistry for maintaining a healthy body, and ways to improve health and virility. Written with a holistic, it emphasizes that a sound mind and sound body go hand in hand. As such, it also stresses aspects relating to mind and morality.

• The sushruta -samhita is considered an important work telling us how advanced surgical science was in olden times. It is difficult to date this volume, although the Mahabharata mentions that sushruta 's father was the sage Vishwamitra. The sushruta -samhita however became widely known and was translated in many languages in Asia and Europe during ninth and tenth centuries A.D.

  1. Ancient Indian doctors had no clear knowledge of the function of the brain, and believed with many ancient peoples that the heart was the seat of intelligence. They realized, however, the importance of the spinal cord and knew the importance of the spinal cord and knew of the existence of the nervous system, though it was not properly understood.

  2. Despite their inaccurate knowledge of physiology , which was by no means inferior to that of most ancient peoples , India evolved a developed empirical surgery. The caesarian section was known , bone-setting reached a high degree of skill, and plastic surgery was developed far beyond anything known elsewhere at the time. Ancient Indian surgeons were expert at the repair of noses, ears, and lips, lost or injured in battle or by judicial mutilation.

  3. The rules of professional behavior laid down in medical texts remind us of those of Hippocrates and are not unworthy of the conscientious doctor of any place or time. We quote part of the sermon, which Charaka instructs a physician to preach to his pupils at a solemn religious ceremony to be performed on the completion of their apprenticeship.

" When you go to the home of a patient you should direct your words, mind, intellect and sense nowhere but to your patient and his treatment..

Nothing that happens in the house of the sick man must be told outside, nor must the patient's condition be told to anyone who might do harm by that knowledge to the patient or to another. ( A L Basham , " The wonder that was India")

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1. The work of the Indian craftsman, however primitive and simple his tools, has been admired for its delicacy and skill. India's spinners and weavers could produce semi-transparent silks and muslins of extreme thinness and which was much in demand in the Roman Empire.
2. In the working of a stone on a large scale India's skill is attested by the enormous monolithic columns of the Mauryan period.. Many of these bear Asoka's inscriptions, but it is not certain that they were made and erected by him; some may have existed before his time. All are made of sandstone from the same quarry at chunar , about twenty five miles south west of varanasi . Some thirty columns have been found in many parts of Northern India, from Sanchi in the South to the Nepalese Tarai in the North. Their sculptured capitals are great as works of art, but evidence of Indian technological achievement the columns are even more significant. Weighing as much as fifty tons and measuring some forty feet, they were carved from a single blocks of stone , given a polish of wonderful hardness and lustre , and often transported many hundreds of miles to their present positions. The process of their manufacture, polishing and transport has not yet been fully explained, and the secret was apparently lost soon after the Mauryan period , when the school of craftsmen who worked the chunar sandstone vanished. Though many fine examples of later stone carving have come down to us, some much more impressive artistically than the Mauryan columns, it is doubtful whether India ever again showed such a complete mastery of the handling of enormous pieces of stone.( A L Basham, " The Wonder that was India" page- 221)
3. The Iron pillar of Meharauli near Delhi, is even more remarkable, though of little artistic value and less immediately impressive than the Mauryan columns. It is a memorial to a king called Chandra , who was probably Chandra Gupta II ( c. 376-415). The pillar was originally erected on a hill near Ambala and brought to delhi. It is over twenty -three feet high , and consists of a single piece of Iron , of a size and weight which could not have been produced by the the best European iron founders until about one hundred years ago. As with the Mauryan columns we have no clear evidence of how it was made , but it must have demanded immense care and labour , and great technical proficiency in preparing and heating the metal . the metallurgical skill of ancient India is further attested by the fact that this pillar , though it has weathered the torrential rains of over 1,500 monsoons , shows no signs of rusting. since the process of oxidization demands a catalyst , it may be the great purity of the metal which has preserved the Iron pillar so long, as another memorial to India's technical skill.