11-20-2005, 10:23 AM
<!--QuoteBegin-->QUOTE<!--QuoteEBegin-->Posted: 27 Aug 2002  Post subject: Re: Ancient (and recent) Indian Mathematics Â
--------------------------------------------------------------------------------
http://www.infinityfoundation.com/sourcebook.htm
ECIT Sourcebook on Indic Contributions in Math and Science
This sourcebook will consist primarily of reprinted articles on Indic contributions in math and science, as well as several new essays to contextualize these works. It will bring together the works of top scholars which are currently scattered thoughout disparate journals, and will thus make them far more accessible to the average reader.
There are two main reasons why this sourcebook is being assembled. First, it is our hope that by highlighting the work of ancient and medieval Indian scientists we might challenge the stereotype that Indian thought is "mystical" and "irrational". Secondly, by pointing out the numerous achievements of Indian scientists, we hope to show that India had a scientific "renaissance" that was at least as important as the European renaissance which followed it, and which, indeed, is deeply indebted to it.
Currently, the following table of contents is proposed for this volume:
1. Editors' Introduction (Subhash Kak)
Section 1: Mathematics
2. D. Gray, 2000. Indic Mathematics etc.
3. Joseph, George Ghevarughese. 1987. "Foundations of Eurocentrism in Mathematics". In Race & Class 28.3, pp. 13-28.
4. A. Seidenberg, 1978. The origin of Mathematics. Archive for History of Exact Sciences 18.4, pp. 301-42.
5. Frits Staal, 1965. Euclid and Panini. Philosophy East and West 15.2, pp. 99-116.
6. Subhash Kak, 2000. Indian binary numbers and the Katapayadi notation. ABORI, 81.
7. Subhash Kak, 1990. The sign for zero. Mankind Quarterly, 30, pp. 199-204.
8. C.-O. Selenius, 1975. Rationale of the chakravala process of Jayadeva and Bhaskara II. Historia Mathematica, 2, pp. 167-184.
9. K.V. Sarma, 1972. Anticipation of modern mathematical discoveries by Kerala astronomers. In A History of the Kerala School of Hindu Astronomy. Hoshiarpur: Vishveshvaranand Institute.
Section 2: Science, General
10. Staal, Frits. 1995. "The Sanskrit of Science". In Journal of Indian Philosophy 23, pp. 73-127.
11. 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.
12. Saroja Bhate and Subhash Kak, 1993. Panini's grammar and computer science. ABORI, 72, pp. 79-94.
Section 3: Astronomy
13. Subhash Kak, 1992. The astronomy of the Vedic altars and the Rgveda. Mankind Quarterly, 33, pp. 43-55.
14. Subhash Kak, 1995. The astronomy of the age of geometric altars. Quarterly Journal of the Royal Astronomical Society, 36, pp. 385-395.
15. Subhash Kak, 1996. Knowledge of planets in the third millennium BC. Quarterly Journal of the Royal Astronomical Society, 37, pp. 709-715.
16. Subhash Kak, 1998. Early theories on the distance to the sun. Indian Journal of History of Science, 33, pp. 93-100.
17. B.N. Narahari Achar, 1998. Enigma of the five-year yuga of Vedanga Jyotisa, Indian Journal of History of Science, 33, pp. 101-109.
18. B.N. Narahari Achar, 2000. On the astronomical basis of the date of Satapatha Brahmana, Indian Journal of History of Science, 35, pp. 1-19.
19. B.L. van der Waerden, 1980. Two treatises on Indian astronomy, Journal for History of Astronomy 11, pp.50-58.
20. K. Ramasubramanian, M.D. Srinivas, M.S. Sriram, 1994. Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion. Current Science, 66, pp. 784-790.
Back to top   Â
Kaushal
BRFite
Joined: 31 Dec 1969
Posts: 481
Posted: 27 Aug 2002  Post subject: Re: Ancient (and recent) Indian Mathematics Â
--------------------------------------------------------------------------------
http://www.infinityfoundation.com/kumargrant.htm
A Brief Discussion on the Contributions of India
"The first nation (to have cultivated science) is India. This is a powerful nation having a large population, and a rich kingdom (possession). India is known for the wisdom of its people. Over many centuries, all the kings of the past have recognized the ability of the Indians in all branches of knowledge," wrote Sa`id al?Andalusi, a leading natural philosopher of the eleventh century Muslim Spain (Salem and Kumar, 1991, p. 11). The emphasis in the above quotation is not on India being "the first nation to cultivate science." It is on the fact that the European scholars, as late as the eleventh-century, thought India as a leader in science and technology. This is in contrast with the modern common perception about India in the Western minds or with the colonial period.
Poetry in Science: As early as 2600 B.C., the Egyptian monarch Khufu erected a great "house of writings," thus inaugurating a tradition that, by the time of Rameses II (ca. 1300 B.C.), yielded a collection numbering at least 20,000 papyrus scrolls, each protected in a cloth or leather cover. During the 300 year period, between the sack of Thebes by the Assyrians and the invasion of Alexander the Great's armies in 332 B.C., practically all of Egypt's libraries were turned to ash and dust. We know, of course, that this great loss was soon compensated-and afterward repeated-by the great library at Alexandria. This was not typical of only Egypt. Similar decimation of the written words occurred elsewhere in China, Europe, and the Middle East. The Qin dynasty in China, the Reformation in Europe, and Kublai Khan invasion of the Middle East remind us clearly and concretely: the written words are fragile.
While the textual riches of Alexandria, China, and Rome were being put to the flame, a wholly different tradition of scientific expression was brought to a peak in India, in a manner that would prove enormously more resilient to the vicissitudes of time and adversity. This was the oral, poetic tradition of Indian thought, whose greatest purveyor in astronomy and mathematics was Aryabhata I (b. 476 A.D.). }Aryabhata I, by almost any account, was the equal in Indian astronomy to what Cladius Ptolemy became to the tradition of Greek science in Islam and late medieval Europe.
Aryabhata I composed a most remarkable work in Sanskrit known as the Aryabhatiya. Its text consists of a mere 121 stanzas, each with several lines in varying metre and rhyme scheme. The whole is divided into a brief introduction (ten verses), followed by three major parts, one on mathematics (Ganita, 33 verses), one on time reckoning and planetary models (Kalakriya, 25 verses), and one on the mathematics of the sphere and its applicability to astronomical calculation (Gola, 50 verses).
There are no numbers anywhere in the composition. Nor are there figures, drawings, or equations. The Aryabhatiya expresses the highly sophisticated mathematics of sine functions, volumetric determinations, calculation of celestial latitudes and motions, and much more, in the form of a poetic code. This code, invented by Aryabhata I (though others like it had existed earlier on) and delineated in the introductory section, uses an alphabetical system of numerical notation. Specific values are assigned to specific letters and letter combinations, such that any value can be expressed in words and recited through the given metric scheme.
Use of a poetic code did mean, however, that transmission of this most valuable text could be achieved orally. This means through memorization and spoken transfer, or more accurately, by this particular text becoming a part of living memory among succeeding generations of scholars. This occurred during much of the medieval era, when many famous libraries and teaching centers were destroyed in India, as elsewhere, by foreign conquest and civil strife. However, many books were reproduced later on from the oral tradition
The image brought to us by the history of Aryabhata I's seminal work is indeed striking: at the very moment when Ptolemy's, Hipparchus', Eudoxus' and Euclid's books darkened the Alexandrian sky with textual ash, the Aryabhatiya was being passed and preserved through the most transient yet durable medium, the spoken and memorized word in the hearts of the Indians.
Mathematics: The present place?value notation system (base 10), used to represent numbers, is Indian in origin. The Mayas (base 20) and the Babylonians (base 60) used place?value systems as well, although their systems were different from that of the Indians. The role of place-value notation is quite important in mathematics and computers. For example, the extreme difficulty of performing mathematical operations in the absence of a place?value notation system caused Greek mathematics and astronomy to suffer. Nicholas Copernicus (A.D. 1473?1543) was forced to use the Hindu numerals for their ingenuity (Rosen, 1978, p. 27): "This [Hindu] numeral notation certainly surpasses every other, whether Greek or Latin, in lending itself to computations with exceptional speed. For this reason I have accepted [it]." Copernicus, in his book, On the Revolutions, used Hindu?numerals for his computations while advocating the heliocentric model of the solar system.
Trigonometry deals with specific functions of angles and their applications to calculations in geometry. It unites the disciplines of arithmetic, algebra, geometry, and astronomy. The Alexandrian Hipparchus (ca. 150 B.C.) and Ptolemy (ca. A.D. 100--170) helped to lay the foundations of trigonometry. With the work of Aryabhata I (ca. A.D. 500) in India, trigonometry began to assume its modern form. For example, consider a circle of unit radius, so that the length of an arc of the circle is a measure of the angle it subtends at the center of the circle. In order to facilitate calculations in geometry, the Greeks tabulated values of the chords of different arcs of a circle. This method was replaced by Indian mathematicians with another system that used the half chord of an arc, known today as the "sine" of an angle.
The origin of the word "sine" in trigonometry can be traced to the Sanskrit language. The Sanskrit word for the chord of an arc comes from an analogy with a "bow string," which is called "samasta?jya." The half?chord of the arc was called "jya?ardha," later shortened to "jya." The Arabs borrowed the concept from India and chose a similarly word "jaib" for the mathematical operation, which linguistically means a "lace fold or pocket" used in clothes.
This Arabic word was literally translated into Latin and called "sinus," meaning curved surface or fold. This Latin word was later metamorphosed into "sine" or "sin" in the English language. (Joseph, 1991; Kumar, 1994) Such etymological knowledge can be found in some dictionaries and encyclopedias but, ironically, only rarely in trigonometry books.
When the Arabs borrowed a concept from the Indians, they usually attributed the source correctly, especially in their early medieval literature. They often kept the same or similar pronunciation for many words, as shown by the previous example. In other instances, they translated the Sanskrit word into Arabic.
For example, the Sanskrit word for zero (a Hindu concept) is sunya, which translates literally into the Arabic word sifr. When the Europeans learned of it, they called it cipher or zephirum in Latin. This was transformed eventually as zero in English.
Natural Sciences: Regarding the earth's motion in his heliocentric model, Aryabhata I, some thousand years before Copernicus, suggested that the earth might be in axial rotation, with the heavens at rest, so that the apparent motion of the stars would be an illusion. In order to explain the apparent motion of the sun, Aryabhata I used the elegant analogy of a boat in a river: "As a man in a boat going forward sees a stationary object moving backward, just so at Lanka (Sri?Lanka) a man sees the stationary asterisms (stars) moving backward in a straight line." (Clark, 1930, Gola 9) The interpretation is that a person standing on the equator of the earth (Sri?Lanka) that rotates toward the east would see the stars (asterisms) moving in a westward direction. Aryabhata I's hypothesis of the earth's rotational motion is clearly explained by the analogy of the boatman as given above. He incorporated the concept of relative motion, many centuries before its more formal discussion by the noted Parisian scholar Nicholas Oresme in the fourteenth century. (Toulmin and Goodfield, 1961).
The ancient Indians realized the common connections between plants, animals and humans. They also realized the role of plants in medicine and the role of animals in the preservation of nature. In view of the importance of trees for humans, unnecessary cutting of a tree was against social and religious codes in India. Asoka (2nd century B.C.), a king in India known for his Maurya?empire and propagation of Buddhism, defined a strict law against the demolition of trees. Similarly, protection of animals led to the movement of vegetarianism. The Hindus went to the extent of worshiping both life forms; Animals and plants are revered along with gods. It is a curious fact that the Hindu concept of transmigration of souls was a common belief among the Greek philosophers and not the Greek society. Some connections between the Greek philosophers and Indian philosophy and the possibility of such interactions will also be explored.
Kanada wrote a book, Vaisesika?Sutra, in which he proposed that matter is made up of small particles, called parmanu (atoms). Kanada's description is based on the impossibility of infinite division of matter: "parmanu is not visible; the non?perception of atoms disappears when they mass together to form a combination of three double?atoms, a combination which does assume visibility." Cyril Bailey compared the Indian description of atoms with the Greek description, and wrote: "It is interesting to realize that an early date Indian philosophers had arrived at an atomic explanation of the universe. The doctrines of this school were expounded by Vaicesika Sutra [VaiÃesika?Sãtra] and interpreted by the aphorisms of Kanada . . . .Kanada works out the idea of their combinations in a detailed system, which reminds us at once of the Pythagoreans and in some respect of modern science, holding that two atoms combined in a binary compound and three of these binaries in a triad which would be a size to be perceptible to the sense." (Bailey 1928, p. 64)
As this section amply demonstrates that India made substantial contributions to science and technology in the ancient and medieval period. However, such contributions are not in the mainstream knowledge in the absence of its inclusion in the academic curricula. This project is a modest effort to bridge such gaps.
Salient Features
Most writings related to Indian scientific contributions are written in specialized or obscure journals on Indology. Most articles are written in isolation with little or no connection/comparison with the western science or with other contemporary cultures. In the absence of a cross-cultural study, it is unlikely that such knowledge will ever be assimilated into the mainstream knowledge of science. For this reason, I plan to write my articles/modules with a cross-cultural emphasis. Such a study will not be in conflict with the mainstream science; it will fill the gaps in our understanding of science. I also plan to publish my articles in the mainstream journals/magazines (examples: Science as Culture, The Physics Teacher, Journal of College Science Teaching, or American Educational Research Journal)
Back to top   Â
Kaushal
BRFite
Joined: 31 Dec 1969
Posts: 481
Posted: 02 Sep 2002  Post subject: Re: Ancient (and recent) Indian Mathematics Â
--------------------------------------------------------------------------------
Kerala School of mathematics
For more on the Kerala School of mathematics, see 'Science and Technology in Ancient india',published by Vijnan Bharati, Mumbai, pp.56.Prominent among the Kerala School were;
Madhava (1340-1425 ce)
Neelakantha Somayaji (1445-1545 ce)
Jyeshtadeva (1520 ce)
Putumana Somayaji (circa 1730)
Raja Sankara Varma
Cross posted from Indiancivilizations
I would like to add contribution of Kerala to Calculus-
On the page 111 of the research paper "The Sanskrit of Science" by Dr.
Frits Stal (Journal of Indian Philosophy, vol. 23, p.73-127, March 1995),
it was mentioned, " The fact remains that Leibnitz, Madhava , Newton,
Nilakantha and others made the same discoveries, the Indians even
earlier,------"
Madhava and Nilakantha were from Kerala.
N. R. Joshi.
If you had to put your finger on why Indian mathematics did not become the engine of engineering and then the economy that their european counterparts became, what would be the reason.
In order for any academic field of endeavor to flourish at an advanced level, there must be peace and relative prosperity in the land and such endeavors must be supported by the state. A good example is the Indian Institute of Science and its successor institutes the IIT's which are churning out good work at a steady rate today. It is AlBiruni who remarked that Ghazni's marauding raids had left all of northwest India in ruin and bereft of the presence of savants in the sciences and mathematics. He specifically makes mention of the fact that many have fled to states beyond the reach of his master. It is no coincidence that Ujjain which was a center of scientific activity during pre-Islamic times ceased to be a center after the medieval era. In 1304 CE Alla ud din in one of his many jihads completely destroyed Ujjain, the center of Indian scholarship. North India never recovered from this as from the earlier destruction of Nalanda by Bakhtyar Khalji in 1198 CE, which destroyed Buddhist scholarship.
The center of gravity of Indian scholarship slowly shifted southwards until the 18th century when the British delivered the coup de grace of English based education, which inflicted the double whammy of lack of continuity as well as cutting of the vast majority of Indians from access to any educational institution.
But the spirit of Indian Mathematics could not be extinguished. It came back with a roar in the person of Ramanujam, the greatest number theorist in all of human history.
Kaushal
<!--QuoteEnd--><!--QuoteEEnd-->
--------------------------------------------------------------------------------
http://www.infinityfoundation.com/sourcebook.htm
ECIT Sourcebook on Indic Contributions in Math and Science
This sourcebook will consist primarily of reprinted articles on Indic contributions in math and science, as well as several new essays to contextualize these works. It will bring together the works of top scholars which are currently scattered thoughout disparate journals, and will thus make them far more accessible to the average reader.
There are two main reasons why this sourcebook is being assembled. First, it is our hope that by highlighting the work of ancient and medieval Indian scientists we might challenge the stereotype that Indian thought is "mystical" and "irrational". Secondly, by pointing out the numerous achievements of Indian scientists, we hope to show that India had a scientific "renaissance" that was at least as important as the European renaissance which followed it, and which, indeed, is deeply indebted to it.
Currently, the following table of contents is proposed for this volume:
1. Editors' Introduction (Subhash Kak)
Section 1: Mathematics
2. D. Gray, 2000. Indic Mathematics etc.
3. Joseph, George Ghevarughese. 1987. "Foundations of Eurocentrism in Mathematics". In Race & Class 28.3, pp. 13-28.
4. A. Seidenberg, 1978. The origin of Mathematics. Archive for History of Exact Sciences 18.4, pp. 301-42.
5. Frits Staal, 1965. Euclid and Panini. Philosophy East and West 15.2, pp. 99-116.
6. Subhash Kak, 2000. Indian binary numbers and the Katapayadi notation. ABORI, 81.
7. Subhash Kak, 1990. The sign for zero. Mankind Quarterly, 30, pp. 199-204.
8. C.-O. Selenius, 1975. Rationale of the chakravala process of Jayadeva and Bhaskara II. Historia Mathematica, 2, pp. 167-184.
9. K.V. Sarma, 1972. Anticipation of modern mathematical discoveries by Kerala astronomers. In A History of the Kerala School of Hindu Astronomy. Hoshiarpur: Vishveshvaranand Institute.
Section 2: Science, General
10. Staal, Frits. 1995. "The Sanskrit of Science". In Journal of Indian Philosophy 23, pp. 73-127.
11. 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.
12. Saroja Bhate and Subhash Kak, 1993. Panini's grammar and computer science. ABORI, 72, pp. 79-94.
Section 3: Astronomy
13. Subhash Kak, 1992. The astronomy of the Vedic altars and the Rgveda. Mankind Quarterly, 33, pp. 43-55.
14. Subhash Kak, 1995. The astronomy of the age of geometric altars. Quarterly Journal of the Royal Astronomical Society, 36, pp. 385-395.
15. Subhash Kak, 1996. Knowledge of planets in the third millennium BC. Quarterly Journal of the Royal Astronomical Society, 37, pp. 709-715.
16. Subhash Kak, 1998. Early theories on the distance to the sun. Indian Journal of History of Science, 33, pp. 93-100.
17. B.N. Narahari Achar, 1998. Enigma of the five-year yuga of Vedanga Jyotisa, Indian Journal of History of Science, 33, pp. 101-109.
18. B.N. Narahari Achar, 2000. On the astronomical basis of the date of Satapatha Brahmana, Indian Journal of History of Science, 35, pp. 1-19.
19. B.L. van der Waerden, 1980. Two treatises on Indian astronomy, Journal for History of Astronomy 11, pp.50-58.
20. K. Ramasubramanian, M.D. Srinivas, M.S. Sriram, 1994. Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion. Current Science, 66, pp. 784-790.
Back to top   Â
Kaushal
BRFite
Joined: 31 Dec 1969
Posts: 481
Posted: 27 Aug 2002  Post subject: Re: Ancient (and recent) Indian Mathematics Â
--------------------------------------------------------------------------------
http://www.infinityfoundation.com/kumargrant.htm
A Brief Discussion on the Contributions of India
"The first nation (to have cultivated science) is India. This is a powerful nation having a large population, and a rich kingdom (possession). India is known for the wisdom of its people. Over many centuries, all the kings of the past have recognized the ability of the Indians in all branches of knowledge," wrote Sa`id al?Andalusi, a leading natural philosopher of the eleventh century Muslim Spain (Salem and Kumar, 1991, p. 11). The emphasis in the above quotation is not on India being "the first nation to cultivate science." It is on the fact that the European scholars, as late as the eleventh-century, thought India as a leader in science and technology. This is in contrast with the modern common perception about India in the Western minds or with the colonial period.
Poetry in Science: As early as 2600 B.C., the Egyptian monarch Khufu erected a great "house of writings," thus inaugurating a tradition that, by the time of Rameses II (ca. 1300 B.C.), yielded a collection numbering at least 20,000 papyrus scrolls, each protected in a cloth or leather cover. During the 300 year period, between the sack of Thebes by the Assyrians and the invasion of Alexander the Great's armies in 332 B.C., practically all of Egypt's libraries were turned to ash and dust. We know, of course, that this great loss was soon compensated-and afterward repeated-by the great library at Alexandria. This was not typical of only Egypt. Similar decimation of the written words occurred elsewhere in China, Europe, and the Middle East. The Qin dynasty in China, the Reformation in Europe, and Kublai Khan invasion of the Middle East remind us clearly and concretely: the written words are fragile.
While the textual riches of Alexandria, China, and Rome were being put to the flame, a wholly different tradition of scientific expression was brought to a peak in India, in a manner that would prove enormously more resilient to the vicissitudes of time and adversity. This was the oral, poetic tradition of Indian thought, whose greatest purveyor in astronomy and mathematics was Aryabhata I (b. 476 A.D.). }Aryabhata I, by almost any account, was the equal in Indian astronomy to what Cladius Ptolemy became to the tradition of Greek science in Islam and late medieval Europe.
Aryabhata I composed a most remarkable work in Sanskrit known as the Aryabhatiya. Its text consists of a mere 121 stanzas, each with several lines in varying metre and rhyme scheme. The whole is divided into a brief introduction (ten verses), followed by three major parts, one on mathematics (Ganita, 33 verses), one on time reckoning and planetary models (Kalakriya, 25 verses), and one on the mathematics of the sphere and its applicability to astronomical calculation (Gola, 50 verses).
There are no numbers anywhere in the composition. Nor are there figures, drawings, or equations. The Aryabhatiya expresses the highly sophisticated mathematics of sine functions, volumetric determinations, calculation of celestial latitudes and motions, and much more, in the form of a poetic code. This code, invented by Aryabhata I (though others like it had existed earlier on) and delineated in the introductory section, uses an alphabetical system of numerical notation. Specific values are assigned to specific letters and letter combinations, such that any value can be expressed in words and recited through the given metric scheme.
Use of a poetic code did mean, however, that transmission of this most valuable text could be achieved orally. This means through memorization and spoken transfer, or more accurately, by this particular text becoming a part of living memory among succeeding generations of scholars. This occurred during much of the medieval era, when many famous libraries and teaching centers were destroyed in India, as elsewhere, by foreign conquest and civil strife. However, many books were reproduced later on from the oral tradition
The image brought to us by the history of Aryabhata I's seminal work is indeed striking: at the very moment when Ptolemy's, Hipparchus', Eudoxus' and Euclid's books darkened the Alexandrian sky with textual ash, the Aryabhatiya was being passed and preserved through the most transient yet durable medium, the spoken and memorized word in the hearts of the Indians.
Mathematics: The present place?value notation system (base 10), used to represent numbers, is Indian in origin. The Mayas (base 20) and the Babylonians (base 60) used place?value systems as well, although their systems were different from that of the Indians. The role of place-value notation is quite important in mathematics and computers. For example, the extreme difficulty of performing mathematical operations in the absence of a place?value notation system caused Greek mathematics and astronomy to suffer. Nicholas Copernicus (A.D. 1473?1543) was forced to use the Hindu numerals for their ingenuity (Rosen, 1978, p. 27): "This [Hindu] numeral notation certainly surpasses every other, whether Greek or Latin, in lending itself to computations with exceptional speed. For this reason I have accepted [it]." Copernicus, in his book, On the Revolutions, used Hindu?numerals for his computations while advocating the heliocentric model of the solar system.
Trigonometry deals with specific functions of angles and their applications to calculations in geometry. It unites the disciplines of arithmetic, algebra, geometry, and astronomy. The Alexandrian Hipparchus (ca. 150 B.C.) and Ptolemy (ca. A.D. 100--170) helped to lay the foundations of trigonometry. With the work of Aryabhata I (ca. A.D. 500) in India, trigonometry began to assume its modern form. For example, consider a circle of unit radius, so that the length of an arc of the circle is a measure of the angle it subtends at the center of the circle. In order to facilitate calculations in geometry, the Greeks tabulated values of the chords of different arcs of a circle. This method was replaced by Indian mathematicians with another system that used the half chord of an arc, known today as the "sine" of an angle.
The origin of the word "sine" in trigonometry can be traced to the Sanskrit language. The Sanskrit word for the chord of an arc comes from an analogy with a "bow string," which is called "samasta?jya." The half?chord of the arc was called "jya?ardha," later shortened to "jya." The Arabs borrowed the concept from India and chose a similarly word "jaib" for the mathematical operation, which linguistically means a "lace fold or pocket" used in clothes.
This Arabic word was literally translated into Latin and called "sinus," meaning curved surface or fold. This Latin word was later metamorphosed into "sine" or "sin" in the English language. (Joseph, 1991; Kumar, 1994) Such etymological knowledge can be found in some dictionaries and encyclopedias but, ironically, only rarely in trigonometry books.
When the Arabs borrowed a concept from the Indians, they usually attributed the source correctly, especially in their early medieval literature. They often kept the same or similar pronunciation for many words, as shown by the previous example. In other instances, they translated the Sanskrit word into Arabic.
For example, the Sanskrit word for zero (a Hindu concept) is sunya, which translates literally into the Arabic word sifr. When the Europeans learned of it, they called it cipher or zephirum in Latin. This was transformed eventually as zero in English.
Natural Sciences: Regarding the earth's motion in his heliocentric model, Aryabhata I, some thousand years before Copernicus, suggested that the earth might be in axial rotation, with the heavens at rest, so that the apparent motion of the stars would be an illusion. In order to explain the apparent motion of the sun, Aryabhata I used the elegant analogy of a boat in a river: "As a man in a boat going forward sees a stationary object moving backward, just so at Lanka (Sri?Lanka) a man sees the stationary asterisms (stars) moving backward in a straight line." (Clark, 1930, Gola 9) The interpretation is that a person standing on the equator of the earth (Sri?Lanka) that rotates toward the east would see the stars (asterisms) moving in a westward direction. Aryabhata I's hypothesis of the earth's rotational motion is clearly explained by the analogy of the boatman as given above. He incorporated the concept of relative motion, many centuries before its more formal discussion by the noted Parisian scholar Nicholas Oresme in the fourteenth century. (Toulmin and Goodfield, 1961).
The ancient Indians realized the common connections between plants, animals and humans. They also realized the role of plants in medicine and the role of animals in the preservation of nature. In view of the importance of trees for humans, unnecessary cutting of a tree was against social and religious codes in India. Asoka (2nd century B.C.), a king in India known for his Maurya?empire and propagation of Buddhism, defined a strict law against the demolition of trees. Similarly, protection of animals led to the movement of vegetarianism. The Hindus went to the extent of worshiping both life forms; Animals and plants are revered along with gods. It is a curious fact that the Hindu concept of transmigration of souls was a common belief among the Greek philosophers and not the Greek society. Some connections between the Greek philosophers and Indian philosophy and the possibility of such interactions will also be explored.
Kanada wrote a book, Vaisesika?Sutra, in which he proposed that matter is made up of small particles, called parmanu (atoms). Kanada's description is based on the impossibility of infinite division of matter: "parmanu is not visible; the non?perception of atoms disappears when they mass together to form a combination of three double?atoms, a combination which does assume visibility." Cyril Bailey compared the Indian description of atoms with the Greek description, and wrote: "It is interesting to realize that an early date Indian philosophers had arrived at an atomic explanation of the universe. The doctrines of this school were expounded by Vaicesika Sutra [VaiÃesika?Sãtra] and interpreted by the aphorisms of Kanada . . . .Kanada works out the idea of their combinations in a detailed system, which reminds us at once of the Pythagoreans and in some respect of modern science, holding that two atoms combined in a binary compound and three of these binaries in a triad which would be a size to be perceptible to the sense." (Bailey 1928, p. 64)
As this section amply demonstrates that India made substantial contributions to science and technology in the ancient and medieval period. However, such contributions are not in the mainstream knowledge in the absence of its inclusion in the academic curricula. This project is a modest effort to bridge such gaps.
Salient Features
Most writings related to Indian scientific contributions are written in specialized or obscure journals on Indology. Most articles are written in isolation with little or no connection/comparison with the western science or with other contemporary cultures. In the absence of a cross-cultural study, it is unlikely that such knowledge will ever be assimilated into the mainstream knowledge of science. For this reason, I plan to write my articles/modules with a cross-cultural emphasis. Such a study will not be in conflict with the mainstream science; it will fill the gaps in our understanding of science. I also plan to publish my articles in the mainstream journals/magazines (examples: Science as Culture, The Physics Teacher, Journal of College Science Teaching, or American Educational Research Journal)
Back to top   Â
Kaushal
BRFite
Joined: 31 Dec 1969
Posts: 481
Posted: 02 Sep 2002  Post subject: Re: Ancient (and recent) Indian Mathematics Â
--------------------------------------------------------------------------------
Kerala School of mathematics
For more on the Kerala School of mathematics, see 'Science and Technology in Ancient india',published by Vijnan Bharati, Mumbai, pp.56.Prominent among the Kerala School were;
Madhava (1340-1425 ce)
Neelakantha Somayaji (1445-1545 ce)
Jyeshtadeva (1520 ce)
Putumana Somayaji (circa 1730)
Raja Sankara Varma
Cross posted from Indiancivilizations
I would like to add contribution of Kerala to Calculus-
On the page 111 of the research paper "The Sanskrit of Science" by Dr.
Frits Stal (Journal of Indian Philosophy, vol. 23, p.73-127, March 1995),
it was mentioned, " The fact remains that Leibnitz, Madhava , Newton,
Nilakantha and others made the same discoveries, the Indians even
earlier,------"
Madhava and Nilakantha were from Kerala.
N. R. Joshi.
If you had to put your finger on why Indian mathematics did not become the engine of engineering and then the economy that their european counterparts became, what would be the reason.
In order for any academic field of endeavor to flourish at an advanced level, there must be peace and relative prosperity in the land and such endeavors must be supported by the state. A good example is the Indian Institute of Science and its successor institutes the IIT's which are churning out good work at a steady rate today. It is AlBiruni who remarked that Ghazni's marauding raids had left all of northwest India in ruin and bereft of the presence of savants in the sciences and mathematics. He specifically makes mention of the fact that many have fled to states beyond the reach of his master. It is no coincidence that Ujjain which was a center of scientific activity during pre-Islamic times ceased to be a center after the medieval era. In 1304 CE Alla ud din in one of his many jihads completely destroyed Ujjain, the center of Indian scholarship. North India never recovered from this as from the earlier destruction of Nalanda by Bakhtyar Khalji in 1198 CE, which destroyed Buddhist scholarship.
The center of gravity of Indian scholarship slowly shifted southwards until the 18th century when the British delivered the coup de grace of English based education, which inflicted the double whammy of lack of continuity as well as cutting of the vast majority of Indians from access to any educational institution.
But the spirit of Indian Mathematics could not be extinguished. It came back with a roar in the person of Ramanujam, the greatest number theorist in all of human history.
Kaushal
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