Lakonian
09-20-2007, 11:24 AM
Under the Gupta kings the city of Pataliputra became the home of scientific studies, especially of astronomy and mathematics, both of which show a definitely Greek impress in accordance with contemporary work in the school of Alexandria. The astronomer Aryabhata (born 476-499) taught here and has left a treatise on astronomy with a section dealing with mathematics. Varahamihisa (505-587) compiled a work known as the Pance-Siddhanlika, a compilation of five standard manuals of astronomy which he abridged. One of these five treatises belongs to the prc-scientific age and is of no scientific value, but the other four show the influence of Alexandrian scholarship: two of them bear the non-Indian names of Romank and Paulisa, the latter giving a table based on Claudius Ptolemy's table of chords. These treatises refer to the Yavanas or Greeks as the great authorities on science. One of the four treatises is thefifth century anonymous Surya Siddhanta or "knowledge by,, the Sun", which became a standard manual for Indian astronomers. Brahmagupta (circ. 628) was an astronomer who lived and worked in Uiiain, where there was an observatory. He wrote an astronomical manual called the Brahma Siddhanta in twenty-one chapters, including special sections on arithmetic (Ganitad'haya) and indeterminate equations (Kutakhadyaka). This work became known to the Arabs during, or a little before, the reign of Harun ar-Rashid and formed the basis of the work which circulated as the Sindhind, a name which represents the Indian Siddhanta.
Under the Sasanid kings of Persia it had been the custom to take and record astronomical observations, no doubt in the first place for astrological purposes, and these records were regularly published as the Zik-i-shatroayar or "royal tables ". The preparation of those tables was not stopped by the Arab conquest, nor were they greatly changed in form, the Persian language was still used and not replaced by Arabic for several centuries, and even then the dates were given with the old Persian months not the months of the Arabic Muslim year. It is known that there was an observatory at Jundi-Shapur, and no doubt observations were taken there as well as in the Persian observatories, but the whole work was and remained in Persian hands. Then, apparently, the Arabs wanted to understand how these observations were taken and recorded d for that purpose the Sindhind was composed and circulated an amongst them. It was the first astronomical manual introduced to the Arabs, and it included not only astronomical-information, but also the mathematical material necessary for its use, mostly dealing with spherical trigonometry.
There is a legend, but it is a dubious one, which puts back the translation of the Sindhind to the reign of al-Mansur, the founder of Baghdad. This legend relates that the Arabs conquered Sind (Scind), the area of the lower Indus, in the days of their expansion after the fall of the Persian monarchy, which has a good historical basis. This conquest did not result in a complete occupation of the country, but certain Arab chieftains were settled there as a kind of military garrison to hold it, and they, very naturally, became semi-independent. When the 'Abbasid revolution took place they seized the opportunity to declare themselves independent and refused to recognize the new dynasty. But al-Mansur would not tolerate this and sent an armed force to chastise them, and after that experience they determined to make their submission and sent an embassy to Baghdad to make terms. With this embassy went an Indian sage named Kankah, who disclosed to the Arabs the wisdom of the Indians, which consisted of a summary of astronomy and the mathematics involved. But Kankah knew no Arabic or Persian, and his speech had to be translated into Persian by an interpreter, and that into Arabic by a second interpreter, a process which rendered the final form of his instruction very involved and obscure. Al-Biruni (d. 1048), the earliest and best Muslim observer of India and Indian things, knew this story but did not believe it and considered it an invention designed to explain why the translation of the Arabic Sindhind was so obscure and unsatisfactory. History knows of no embassy sent from Sind to al-Mansur. The probability is that the work was an Arabic translation of a Persian version of the Siddhanta already in use in Jundi-Shapur. In any case its contents are not a collection of notes of the discourse of any sage, but a translation, or rather paraphrase, of the standard Indian manual, the revised Siddhanta of Brahmagupta. There may be this much truth in the story, that the Siddhanta passed through two translations on its way to the Arabs, or possibly three, from Indian to Persian, possibly thence into Syriac, finally into Arabic.
The mathematics and astronomy which the Arabs learned from their Indian teachers through a Persian medium were of Greek origin, passed from Alexandria to North-West India. But it does not seem that the actual Greek authorities circulated in India, their teaching was assimilated and restated by Indian scientists, who developed and made material contributions to the material which passed through their hands, and rendered it more flexible by the use of a decimal notation and a greatly increased use of symbols. This can be estimated by noting the work of Aryabhata. It appears from al-Biruni that there were two scientists bearing this name (al-Biruni, India, ii, 305, 327). The elder of these seems to have died about A.D. 500, the date of the younger one is unknown, nor can we always distinguish which of the two is meant. The elder Aryabhata worked at Pataliputra, not at Uiiain. He produced several works, the Gitika, which was a collection of astronomical tables, the Aryashtasata, which includes a treatise on arithmetic known as the Ganita, and a treatise on the geometry of the sphere the necessary basis of astronon-dcal work known as the Gola. He solved quadratic equations,. already anticipated by Diophantus who, however, recognized only one root, even where both are positive, and had been already suggested by Heron. He attempted indeterminate linear equations, already anticipated by Hypsicles, and gives one of the earliest attempts at the general solution of such equations by means of continued fractions. He sums up an arithmetical series after the pth term in a way which may be expressed-
S = n(a+((n-1)/2 + p))d
He gives rules for determining the area of plane figures, but often expresses himself very imperfectly, as "the area produced by a trilateral is the product of the perpendicular which bisects the base and half the base". He gives the area of the sphere as ðr2?(ðr2), which makes ð=16/9, perhaps error for Ahmes' (16/9)2. For the value of he says, "add four to one hundred, multiply by eight, add sixty-two thousand the result is the approximate value of the circumference when the diameter is twenty thousand." This makes ð=62832/20000 or 3.1418.
In his astronomical tables he includes a brief table of sines and rules for finding them. In all this there are traces of Greek teaching, and that appears also in his terminology, as jamitra=äéÜìå“ñïò, kendra=êÝõ“ñïõ, and drama=äñáöìÞ. His work goes farther than that of the Greeks because, like other Indian scientists, he makes a freer use of algebraic expressions, which were rather tentatively introduced by diophantus, and employs the far more convenient Hindu numerals.
Brahmagupta (circ. 628) worked in the Ujain observatory. He was the author of the Brahma-Siddhanta "Brahma's revised Siddhanta", which was the basis of the Arabic Sindhind. This work contains chapters on arithmetic and a treatment of indeterminate equations. In the arithmetic he deals with integers, fractions, progression, barter, rule of three, simple interest, mensuration of plane figures, volumes, and "shadow reckoning" or use of the sun dial. His rules for areas are often defective: thus for an equilateral triangle with side 12 he gives 5x13= 65; for a triangle with sides 13, 14, 15 he gives 7x½x (13+15) which is 96. His formula for the area of a quadrilateral with sides a, b, c, d, is v((s-a)(s-b)(s-c)(s-d)), where s = ½(a+b+c+d), but this is true only for cyclic quadrilaterals. His rule is expressed thus, "Half the sum of the sides set down four times and severally diminished by the sides, being multiplied together, the square root of the product is the exact area." He takes ð as 3 for practical purposes, or v10 as its exact value. He deals with quadratic equations of the type x2+px-q=0, taking x=v(p2-49-p)/2 which gives one root correctly. More important is his application of algebra to astronomy in the Kutakhdyaka, the first instance of such an application being made. He considers simultaneous equations of the first degree, calling their unknowns "colours". Considering the solution of ax-by=c, he gives x = ± cq - bt, and y = ±cp-at. This had been already considered by Aryabhata, who, however, did not solve it, now Brahmagupta gives a solution. These formulae assume that t = zero or any integer and that p/9 is the penultimate convergent of 9/6. For the right-angled triangle he gives two sets of values, 2mn, m2-n2, m2-n, and vm, ½(m/11-n), ½(m/11+n), in which he probably draws from Greek sources. For such treatment it is obvious that Indian mathematics of the period when there was a regular sea route in use between Alexandria and Ujain were based on Alexandrian Greek teaching.
As Arab astronomy began with a continuation of the work in progress in the Persian observatories, which work was rendered possible only by the use of Indian mathematics, it seems fairly certain that the Arabs must have used this Greek science which came throu h an Indian medium, and was transmitted from the Indian scientists by Persian astronomers and mathematicians, although the Persian books which supplied the Arabs with this knowledge are no longer available. It is said that when tlie Arabs found themselves unable to understand the Almajest Ja'far ibn Yahya the Barmakid at once knew the required remedy to be a knowledge of the text of Euclid and Claudius Ptolemy, material at that time not yet accessible in Arabic. If this statement can be treated as reliable it suggests that he, a Persian of Persian education, was familiar with the needed material, though a Persian version, or for that matter an Indian one, of those two authorities is unknown. It is not necessary to prove that translations of the Greek scientists were actually made in Hindu or Persian, it is sufficiently clear that their teaching was known and used
SIGAR O ELLINAS IMON, EKTOS ALON OU KITHOMEN!! lol:laugh:
Under the Sasanid kings of Persia it had been the custom to take and record astronomical observations, no doubt in the first place for astrological purposes, and these records were regularly published as the Zik-i-shatroayar or "royal tables ". The preparation of those tables was not stopped by the Arab conquest, nor were they greatly changed in form, the Persian language was still used and not replaced by Arabic for several centuries, and even then the dates were given with the old Persian months not the months of the Arabic Muslim year. It is known that there was an observatory at Jundi-Shapur, and no doubt observations were taken there as well as in the Persian observatories, but the whole work was and remained in Persian hands. Then, apparently, the Arabs wanted to understand how these observations were taken and recorded d for that purpose the Sindhind was composed and circulated an amongst them. It was the first astronomical manual introduced to the Arabs, and it included not only astronomical-information, but also the mathematical material necessary for its use, mostly dealing with spherical trigonometry.
There is a legend, but it is a dubious one, which puts back the translation of the Sindhind to the reign of al-Mansur, the founder of Baghdad. This legend relates that the Arabs conquered Sind (Scind), the area of the lower Indus, in the days of their expansion after the fall of the Persian monarchy, which has a good historical basis. This conquest did not result in a complete occupation of the country, but certain Arab chieftains were settled there as a kind of military garrison to hold it, and they, very naturally, became semi-independent. When the 'Abbasid revolution took place they seized the opportunity to declare themselves independent and refused to recognize the new dynasty. But al-Mansur would not tolerate this and sent an armed force to chastise them, and after that experience they determined to make their submission and sent an embassy to Baghdad to make terms. With this embassy went an Indian sage named Kankah, who disclosed to the Arabs the wisdom of the Indians, which consisted of a summary of astronomy and the mathematics involved. But Kankah knew no Arabic or Persian, and his speech had to be translated into Persian by an interpreter, and that into Arabic by a second interpreter, a process which rendered the final form of his instruction very involved and obscure. Al-Biruni (d. 1048), the earliest and best Muslim observer of India and Indian things, knew this story but did not believe it and considered it an invention designed to explain why the translation of the Arabic Sindhind was so obscure and unsatisfactory. History knows of no embassy sent from Sind to al-Mansur. The probability is that the work was an Arabic translation of a Persian version of the Siddhanta already in use in Jundi-Shapur. In any case its contents are not a collection of notes of the discourse of any sage, but a translation, or rather paraphrase, of the standard Indian manual, the revised Siddhanta of Brahmagupta. There may be this much truth in the story, that the Siddhanta passed through two translations on its way to the Arabs, or possibly three, from Indian to Persian, possibly thence into Syriac, finally into Arabic.
The mathematics and astronomy which the Arabs learned from their Indian teachers through a Persian medium were of Greek origin, passed from Alexandria to North-West India. But it does not seem that the actual Greek authorities circulated in India, their teaching was assimilated and restated by Indian scientists, who developed and made material contributions to the material which passed through their hands, and rendered it more flexible by the use of a decimal notation and a greatly increased use of symbols. This can be estimated by noting the work of Aryabhata. It appears from al-Biruni that there were two scientists bearing this name (al-Biruni, India, ii, 305, 327). The elder of these seems to have died about A.D. 500, the date of the younger one is unknown, nor can we always distinguish which of the two is meant. The elder Aryabhata worked at Pataliputra, not at Uiiain. He produced several works, the Gitika, which was a collection of astronomical tables, the Aryashtasata, which includes a treatise on arithmetic known as the Ganita, and a treatise on the geometry of the sphere the necessary basis of astronon-dcal work known as the Gola. He solved quadratic equations,. already anticipated by Diophantus who, however, recognized only one root, even where both are positive, and had been already suggested by Heron. He attempted indeterminate linear equations, already anticipated by Hypsicles, and gives one of the earliest attempts at the general solution of such equations by means of continued fractions. He sums up an arithmetical series after the pth term in a way which may be expressed-
S = n(a+((n-1)/2 + p))d
He gives rules for determining the area of plane figures, but often expresses himself very imperfectly, as "the area produced by a trilateral is the product of the perpendicular which bisects the base and half the base". He gives the area of the sphere as ðr2?(ðr2), which makes ð=16/9, perhaps error for Ahmes' (16/9)2. For the value of he says, "add four to one hundred, multiply by eight, add sixty-two thousand the result is the approximate value of the circumference when the diameter is twenty thousand." This makes ð=62832/20000 or 3.1418.
In his astronomical tables he includes a brief table of sines and rules for finding them. In all this there are traces of Greek teaching, and that appears also in his terminology, as jamitra=äéÜìå“ñïò, kendra=êÝõ“ñïõ, and drama=äñáöìÞ. His work goes farther than that of the Greeks because, like other Indian scientists, he makes a freer use of algebraic expressions, which were rather tentatively introduced by diophantus, and employs the far more convenient Hindu numerals.
Brahmagupta (circ. 628) worked in the Ujain observatory. He was the author of the Brahma-Siddhanta "Brahma's revised Siddhanta", which was the basis of the Arabic Sindhind. This work contains chapters on arithmetic and a treatment of indeterminate equations. In the arithmetic he deals with integers, fractions, progression, barter, rule of three, simple interest, mensuration of plane figures, volumes, and "shadow reckoning" or use of the sun dial. His rules for areas are often defective: thus for an equilateral triangle with side 12 he gives 5x13= 65; for a triangle with sides 13, 14, 15 he gives 7x½x (13+15) which is 96. His formula for the area of a quadrilateral with sides a, b, c, d, is v((s-a)(s-b)(s-c)(s-d)), where s = ½(a+b+c+d), but this is true only for cyclic quadrilaterals. His rule is expressed thus, "Half the sum of the sides set down four times and severally diminished by the sides, being multiplied together, the square root of the product is the exact area." He takes ð as 3 for practical purposes, or v10 as its exact value. He deals with quadratic equations of the type x2+px-q=0, taking x=v(p2-49-p)/2 which gives one root correctly. More important is his application of algebra to astronomy in the Kutakhdyaka, the first instance of such an application being made. He considers simultaneous equations of the first degree, calling their unknowns "colours". Considering the solution of ax-by=c, he gives x = ± cq - bt, and y = ±cp-at. This had been already considered by Aryabhata, who, however, did not solve it, now Brahmagupta gives a solution. These formulae assume that t = zero or any integer and that p/9 is the penultimate convergent of 9/6. For the right-angled triangle he gives two sets of values, 2mn, m2-n2, m2-n, and vm, ½(m/11-n), ½(m/11+n), in which he probably draws from Greek sources. For such treatment it is obvious that Indian mathematics of the period when there was a regular sea route in use between Alexandria and Ujain were based on Alexandrian Greek teaching.
As Arab astronomy began with a continuation of the work in progress in the Persian observatories, which work was rendered possible only by the use of Indian mathematics, it seems fairly certain that the Arabs must have used this Greek science which came throu h an Indian medium, and was transmitted from the Indian scientists by Persian astronomers and mathematicians, although the Persian books which supplied the Arabs with this knowledge are no longer available. It is said that when tlie Arabs found themselves unable to understand the Almajest Ja'far ibn Yahya the Barmakid at once knew the required remedy to be a knowledge of the text of Euclid and Claudius Ptolemy, material at that time not yet accessible in Arabic. If this statement can be treated as reliable it suggests that he, a Persian of Persian education, was familiar with the needed material, though a Persian version, or for that matter an Indian one, of those two authorities is unknown. It is not necessary to prove that translations of the Greek scientists were actually made in Hindu or Persian, it is sufficiently clear that their teaching was known and used
SIGAR O ELLINAS IMON, EKTOS ALON OU KITHOMEN!! lol:laugh: