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Indian Inventions, Discoveries and Other Contributions   [Copy link] 中文

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Post time 2007-6-19 07:00:40 |Display all floors
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;
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Post time 2007-6-19 07:01:53 |Display all floors
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.
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Post time 2007-6-19 07:02:40 |Display all floors
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.
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Science and technology in ancient India covered many major branches of human knowledge and activities, including mathematics, astronomy and physics, metallurgy, medical science and surgery, fine arts, mechanical and production technology, civil engineering and architecture, shipbuilding and navigation, sports and games.

According to the 19th century British historian, Grant Duff:

    "
Many of the advances in the sciences that we consider today to have been made in Europe were in fact made in India centuries ago."
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Post time 2007-6-19 07:13:16 |Display all floors
Astronomy

Ancient India’s contributions to astronomy are well known and documented. The earliest references to astronomy are found in the Rig Veda, which are dated to be from 4500 BC. By 500 AD, ancient Indian astronomy emerged as an important part of Indian studies and its affect is seen in several treatises of that period. In some instances, astronomical principles were borrowed to explain matters pertaining to astrology (called Jyotisha in India), like the casting of a horoscope. Apart from this link of astronomy to astrology in ancient India, the science of astronomy continued to develop independently and culminated in original findings such as:

    * The calculation of occurrences of eclipses
    * Calculation of Earth’s circumference
    * Theorizing about gravity
    * Determining that the Sun is a star
    * Determining the number of planets in the Solar System
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Post time 2007-6-19 07:14:35 |Display all floors
Vedic astronomy

There are astronomical references of chronological significance in the Vedas. Some Vedic notices mark the beginning of the year and that of the vernal equinox in Orion; this was the case around 4500 BC. Fire altars, with astronomical basis, have been found in the third millennium cities of India. The texts that describe their designs are conservatively dated to the first millennium BC, but their contents appear to be much older.

A text on Hindu astronomy was written by Lagadha.

The earliest concept of a heliocentric model of the solar system, in which the Sun that is at the centre of the solar system and the Earth that is orbiting it, is found in several Vedic Sanskrit texts written in ancient India.

The Aitareya Brahmana (2.7) (c. 9th–8th century BC) states: "The Sun never sets nor rises. When people think the sun is setting, it is not so; they are mistaken." This indicates that the Sun is stationary (hence the Earth is moving around it), which is elaborated in a later commentary Vishnu Purana (2.8) (c. 1st century), which states: "The sun is stationed for all time, in the middle of the day. [...] Of the sun, which is always in one and the same place, there is neither setting nor rising."

Yajnavalkya (c. 3rd millennium BC) recognized that the Earth was round and believed that the Sun was "the centre of the spheres" as described in the Vedas at the time. His astronomical text Shatapatha Brahmana (8.7.3.10) stated: "The sun strings these worlds - the earth, the planets, the atmosphere - to himself on a thread." He recognized that the Sun was much larger than the Earth, which would have influenced this early heliocentric concept. He also accurately measured the relative distances of the Sun and the Moon from the Earth as 108 times the diameters of these heavenly bodies, almost close to the modern measurements of 107.6 for the Sun and 110.6 for the Moon. (However this could be coincidence, since 108 was a sacred Hindu number.)

Based on his heliocentric model, Yajnavalkya proposed a 95-year cycle to synchronize the motions of the Sun and the Moon, which gives the average length of the tropical year as 365.24675 days, which is only six minutes longer than the modern value of 365.24220 days. This estimate for the length of the tropical year remained the most accurate anywhere in the world for over a thousand years.

There is an old Sanskrit shloka (couplet) which also states "Sarva Dishanaam, Suryaha, Suryaha, Suryaha" which means that there are suns in all directions. This couplet which describes the night sky as full of suns, indicates that in ancient times Indian astronomers had arrived at the important discovery that the stars visible at night are similar to the Sun visible during day time. In other words, it was recognized that the sun is also a star, though the nearest one. This understanding is demonstrated in another Sloka which says that when one sun sinks below the horizon, a thousand suns take its place.
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Post time 2007-6-20 03:36:47 |Display all floors
Grandiose time scales

Hinduism’s understanding of time is as grandiose as time itself. While most cultures base their cosmologies on familiar units such as few hundreds or thousands of years, the Hindu concept of time embraces billions and trillions of years. The Puranas describe time units from the infinitesimal truti, lasting 1/1,000,0000 of a second to a mahamantavara of 311 trillion years. Hindu sages describe time as cyclic, an endless procession of creation, preservation and dissolution. Scientists such as Carl Sagan have expressed amazement at the accuracy of space and time descriptions given by the ancient rishis and saints, who fathomed the secrets of the universe through their mystically awakened senses.

(source: Hinduism Today April/May/June 2007 p. 14).
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